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1303_40 | The Park Board proceeded to commission an independent report from US wildlife veterinarian and scientist Dr. Joseph Gaydos in which he examined the aquarium's animal care standards, accreditation and research and compared it to similar facilities in North America. In his report he found that the Vancouver Aquarium "either meets or exceeds North American industry standards". He also concluded that the aquarium had "an active research department that seems to make good use of studying captive cetaceans, not only for being able to provide better care and understanding of captive animals, but to a greater extent [...] to benefit our understanding and conservation of cetaceans in the wild." Dr. Gaydos also made two recommendations. He recommended that the Park Board conduct "a large-scale scientific study on the welfare of captive housed cetaceans" as a way of assessing "the complex societal issue of captive cetaceans". He also suggested that the Park Board require the aquarium to release |
1303_41 | an annual report on the state of its cetaceans. |
1303_42 | In the media, the Gaydos report was widely received as a positive review of the aquarium's practices and the Vancouver Aquarium also received support from prominent philanthropists and politicians, as well as four former Vancouver mayors. The Monterey Bay Aquarium and the Aquarium of the Pacific in Long Beach, California also spoke out in favour of the Vancouver Aquarium's cetacean program, citing its importance for scientific research. |
1303_43 | After a long public debate that ended with the presentation of the Gaydos report and two days of public hearings, the Vancouver Park Board announced in August 2014 that it intended to enact a by-law to ban breeding of cetaceans at the Vancouver Aquarium, and it tasked its staff with drafting an amendment to the existing by-law regulating cetaceans at the aquarium. Vancouver Aquarium CEO Dr. John Nightingale criticized the decision in a public letter, stating that the decision "was not based on the facts or science presented" and that it did not take into consideration "testimony from dozens of the world’s scientific community, including experts in animal welfare and animal cognition.".
Following a defeat in the elections to Park Board as part of the 2014 civic elections, a majority of the Park Board commissioners, in their last session, voted against enacting an amendment to the by-law. |
1303_44 | Federal law banning captivity of cetaceans
In 2019, the Ending the Captivity of Whales and Dolphins Act became law in Canada. Two facilities would be affected, Marineland of Canada and the Vancouver Aquarium. When passed in June 2019, Marineland was reported to have 61 cetaceans, while the Vancouver Aquarium had just one dolphin remaining. The law has a grandfather clause, permitting those cetaceans already in captivity to remain where they are, but breeding and further acquisition of cetaceans is prohibited, subject to limited exceptions.
In popular culture
The Vancouver Aquarium was featured frequently in the 1980s Canadian series, Danger Bay, which followed the day to day exploits of the Roberts family, led by Grant "Doc" Roberts, a marine veterinarian and his two children, Nicole and Jonah. |
1303_45 | A YouTube video featuring two sea otters "holding hands" was recorded at the Vancouver Aquarium. The two sea otters are Nyac and Milo. Nyac died on September 23, 2008. She was one of the last surviving sea otters of the 1989 Exxon Valdez oil spill. The video has been viewed over 19 million times on YouTube. As a result, the Vancouver Aquarium created a live sea otter cam on their website. The YouTube video was originally recorded by Cynthia Holmes. Milo died on January 12, 2012.
The Vancouver Aquarium was also featured in the family film Andre (1994), and romantic comedy Good Luck Chuck (2007), as Cam's workplace. Television movie The Suite Life Movie (2011) used the aquarium as the research firm where Cody Martin interns.
On September 5, 2008, Hayden Panettiere appeared on the Late Show with David Letterman and talked about her visit with the rescue dolphins at the Vancouver Aquarium. |
1303_46 | The song "Baby Beluga" by Raffi was inspired by Kavna, a beluga that he saw while visiting the Vancouver Aquarium.
References
Bibliography
This is a history of the aquarium as told by the founding and current presidents of the aquarium.
Waters is a magazine published by Canada Wide Media Limited for the official members of the Vancouver Aquarium. It is published three times a year.
External links
Stanley Park
Buildings and structures in Vancouver
Aquaria in Canada
Tourist attractions in Vancouver
Wildlife rehabilitation
Marine mammal rehabilitation and conservation centers |
1304_0 | The 1968 science fiction film 2001: A Space Odyssey featured numerous fictional future technologies, which have proven prescient in light of subsequent developments around the world. Before the film's production began, director Stanley Kubrick sought technical advice from over fifty organizations,
and a number of them submitted their ideas to Kubrick of what kind of products might be seen in a movie set in the year 2001. The film is also praised for its accurate portrayal of spaceflight and vacuum.
Science |
1304_1 | Accuracy
2001 is, according to four NASA engineers who based their nuclear-propulsion spacecraft design in part on the film's Discovery One, "perhaps the most thoroughly and accurately researched film in screen history with respect to aerospace engineering". Several technical advisers were hired for 2001, some of whom were recommended by co-screenwriter Arthur C. Clarke, who himself had a background in aerospace. Advisors included Marshall Spaceflight Center engineer Frederick I. Ordway III, who worked on the film for two years, and I. J. Good, whom Kubrick consulted on supercomputers because of Good's authorship of treatises such as "Speculations Concerning the First Ultraintelligent Machine" and "Logic of Man and Machine".
Dr. Marvin Minsky, of MIT, was the main artificial intelligence adviser for the film. |
1304_2 | 2001 accurately presents outer space as not allowing the propagation of sound, in sharp contrast to other films with space scenes in which explosions or sounds of passing spacecraft are heard. 2001 portrayal of weightlessness in spaceships and outer space is also more realistic. Tracking shots inside the rotating wheel providing artificial gravity contrast with the weightlessness outside the wheel during the repair and Hal disconnection scenes. (Scenes of the astronauts in the Discovery pod bay, along with earlier scenes involving shuttle flight attendants, depict walking in zero-gravity with the help of velcro-equipped shoes labeled "Grip Shoes"). Other aspects that contribute to the film's realism are the depiction of the time delay in conversations between the astronauts and Earth due to the extreme distance between the two (which the BBC announcer explains have been edited out of the broadcast), the attention to small details such as the sound of breathing inside the spacesuits, |
1304_3 | the conflicting spatial orientation of astronauts inside a zero-gravity spaceship, and the enormous size of Jupiter in relation to the spaceship. |
1304_4 | The general approach to how space travel is engineered is highly accurate; in particular, the design of the ships was based on actual engineering considerations rather than attempts to look aesthetically "futuristic". Many other science-fiction films give spacecraft an aerodynamic shape, which is superfluous in outer space (except for craft such as the Pan Am shuttle that are designed to function both in atmosphere and in space). Kubrick's science advisor, Frederick Ordway, notes that in designing the spacecraft "We insisted on knowing the purpose and functioning of each assembly and component, down to the logical labeling of individual buttons and the presentation on screens of plausible operating, diagnostic and other data." Onboard equipment and panels on various spacecraft have specific purposes such as alarm, communications, condition display, docking, diagnostic, and navigation, the designs of which relied heavily on NASA's input. Aerospace specialists were also consulted on the |
1304_5 | design of the spacesuits and space helmets. The space dock at Moon base Clavius shows multiple underground layers which could sustain high levels of air pressure typical of Earth. The lunar craft design takes into account the lower gravity and lighting conditions on the Moon. The Jupiter-bound Discovery is meant to be powered by a nuclear reactor at its rear, separated from the crew area at the front by hundreds of feet of fuel storage compartments. Although difficult to be recognized as such, actual nuclear reactor control panel displays appear in the astronaut's control area. |
1304_6 | The suspended animation of three of the astronauts on board is accurately portrayed as worked out by consulting medical authorities. Such hibernation would likely be necessary to conserve resources on a flight of this kind, as Clarke's novelization implies.
A great deal of effort was made to get the look of the lunar landscape right, based on detailed lunar photographs taken from observatory telescopes. The depiction of early hominids was based on the writings of anthropologists such as Louis Leakey.
Inaccuracy
The film is scientifically inaccurate in minor but revealing details; some due to the technical difficulty involved in producing a realistic effect, and others simply being examples of artistic license. |
1304_7 | The appearance of outer space is problematic, both in terms of lighting and the alignment of astronomical bodies. In the vacuum of outer space, stars do not twinkle, and light does not become diffuse and scattered as it does in air. The side of the Discovery spacecraft unlit by the sun, for example, would appear virtually pitch-black in space. The stars would not appear to move in relation to Discovery as it traveled towards Jupiter, unless it was changing direction. Proportionally, the Sun, Moon and Earth would not visually line up at the size ratios shown in the opening shot, nor would the Galilean moons of Jupiter align as in the shot just before Bowman enters the Star Gate. Kubrick himself was aware of this latter point. (Due to the perfect Laplace resonance of the orbits of the four large moons of Jupiter, the first three never align, and the third moon, Ganymede, is always exactly 90 degrees away from the other two whenever the two innermost moons are in perfect alignment.). |
1304_8 | Similarly, during the scene in the Dawn of Man, where the Sun is seen above the monolith, a crescent moon is depicted close by in the sky. During this phase of the lunar cycle the Moon would be "new" and therefore be invisible. Finally, the edge of Earth appears sharp in the movie, when in reality it is slightly diffuse due to the scattering of the sunlight by the atmosphere, as is seen in many photos of Earth taken from space since the film's release. |
1304_9 | The sequence in which Bowman re-enters Discovery shows him holding his breath just before ejecting from the pod into the emergency airlock. Doing this before exposure to a vacuum—instead of exhaling—would, in reality, rupture the lungs. In an interview on the 2007 DVD release of the film, Clarke states that had he been on the set the day they filmed this, he would have caught this error. In the same scene, the blown pod hatch simply and inexplicably vanishes while concealed behind a puff of smoke. |
1304_10 | When spacecraft land on the Moon in the film, dust is shown billowing as it would in air, not moving in a sheet as it would in the vacuum of the Lunar surface, as can be seen in Apollo Moon landing footage. While on the Moon, all actors move as if in normal Earth gravity, not as they would in the 1/6 gravity of the Moon. Similarly, the behavior of Dave and Frank in the weightless pod bay is not fully consistent with a zero-G environment. Although the astronauts are wearing zero-G 'grip shoes' in order to walk normally, they are oddly leaning on the table while testing the AE-35 unit as if held down by gravity. Finally, in an environment with a radius as small as the main quarters, the simulated gravity would vary significantly from the center of the crew quarters to the 'floor', even varying between feet, waist, and head. The rotation speed of the crew quarters was meant to be only fast enough to generate an approximation of the Moon's gravity, not that of the Earth. However, Clarke |
1304_11 | felt this was enough to prevent the physical atrophy that would result from complete weightlessness. |
1304_12 | The first two appearances of the monolith, one on Earth and one on the Moon, conclude with the Sun at its zenith over the top of the monolith. While this could happen in an African veldt anywhere between the Tropic of Cancer and the Tropic of Capricorn, it could not happen anywhere near the crater Tycho (where the monolith is found) as it is 45 degrees south of the lunar equator. Also implausible is the Sun reaching its zenith so soon after a lunar sunrise, and the appearance of a crescent Earth near the Sun is in complete discontinuity with all previous appearances of Earth, whose position from any spot on the Moon varies only slightly due to libration. |
1304_13 | During Floyd's approach to the space station, parts of the spinning wheel appear to be under construction, consisting of nothing more than bare internal structure. Geophysicist Dr. David Stephenson in the Canadian TV documentary 2001 and Beyond notes that "Every engineer that saw it [the space station] had a fit. You do not spin on a wheel that is not fully built. You have to finish it before you spin it or else you have real problems". |
1304_14 | There are other problems that might be more appropriately described as continuity errors, such as the back-and-forth horizontal switching of Earth's lit side when viewed from Clavius, and the schematic of the space station on the Pan Am spaceplane's monitors continuing to rotate after the plane has synchronized its motion with the station. The latter is due to the position readout actually being a rear-projected film shown in a continuous loop, and being out of sync with other visual elements. The direction of the rotation of the Earth's image outside the space station window is clockwise when Floyd is greeted by a receptionist, but counterclockwise when he phones his daughter. |
1304_15 | Imagining the future
Over fifty organizations contributed technical advice to the production, and a number of them submitted their ideas to Kubrick of what kind of products might be seen in a movie set in the year 2001. Much was made by MGM's publicity department of the film's realism, claiming in a 1968 brochure that "Everything in 2001: A Space Odyssey can happen within the next three decades, and...most of the picture will happen by the beginning of the next millennium." Although the predictions central to the plot—colonization of the Moon, manned interplanetary travel and artificial intelligence—did not materialize by that date, some of the film's other futuristic elements have indeed been realized.
Depiction of computers |
1304_16 | As the central character of the "Jupiter Mission" segment of the film, HAL was shown by Kubrick to have as much intelligence as human beings, possibly more, while sharing their same "emotional potentialities". Kubrick agreed with computer theorists who believed that highly intelligent computers that can learn by experience will inevitably develop emotions such as fear, love, hate, and envy. Such a machine, he said, would eventually manifest human mental disorders as well, such as a nervous breakdown—as Hal did in the film. |
1304_17 | Clarke noted that, contrary to popular rumor, it was a complete coincidence that each of the letters of Hal's name immediately preceded those of IBM in the alphabet. The meaning of HAL has been given both as "Heuristically programmed ALgorithmic computer" and as "Heuristic ALgorithmic computer". The former appears in Clarke's novel of 2001 and the latter in his sequel novel 2010. In computer science, a heuristic is a programmable procedure not necessarily based on fixed rules, producing informed guesses often using trial-and-error. The results can be false such as in predictions of stock market, sports scores, or the weather. Sometimes this can entail selecting on-the-fly one of several methods to solve a problem based on previous experience. On the other hand, an algorithm is a programmable procedure that produces reproducible results using invariant established methods (such as computing square roots). |
1304_18 | A heuristic approach that usually works within a tolerable margin of error may be preferred over a perfect algorithm that requires a long time to run. |
1304_19 | During Apple and Samsung's patent war over consumer electronics design, in 2011 Samsung used a still image from the scene in which two astronauts are eating at a table with what appear to be tablet computers as an exhibit to counter Apple's patent claiming the original abstract design of tablet computers.
Common 21st-century computer technology not depicted in the film include keyboards, mice, mobile phones, touch screens, interfaces with windows/menus/icons. Although there are devices that resemble tablet computers, they are only used in the film as portable video screens.
Depiction of spacecraft |
1304_20 | All of the vehicles in 2001 were designed with extreme care in order for the small-scale models as well as full-scale interiors to appear realistic. The modeling team was led by Kubrick's two hirees from NASA, science advisor Fred Ordway and production designer Harry Lange, along with Anthony Masters who was responsible for turning Lange's 2-D sketches into models. Ordway and Lange insisted on knowing "the purpose and functioning of each assembly and component, down to the labeling of individual buttons and the presentation on screens of plausible operating, diagnostic and other data." Kubrick's team of thirty-five designers was often frustrated by script changes done after designs for various spacecraft had been created. Douglas Trumbull, chief special effects supervisor, writes "One of the most serious problems that plagued us throughout the production was simply keeping track of all ideas, shots, and changes and constantly re-evaluating and updating designs, storyboards, and the |
1304_21 | script itself. To handle all of this....a "control room"...was used to keep track of all progress on the film." Ordway (who worked on designing the station and the five principal space vehicles) has noted that U.S. industry had problems satisfying Kubrick with its equipment suggestions, while design aspects of the vehicles had to be updated often to accommodate rapid screenplay changes, one crew member resigning over an unspecified related issue. Eventually, conflicting ideas of what Kubrick had in mind, what Clarke was writing, and equipment and vehicular realities emerging from Ordway, Lange, Masters, and construction supervisor Dick Frift and his team were resolved, and coalesced into final designs and construction of the spacecraft before filming began in December 1965. |
1304_22 | Other technologies |
1304_23 | One futuristic device shown in the film already under development when the film was released in 1968 was voice-print identification; the first prototype was released in 1976. A credible prototype of a chess-playing computer already existed in 1968, even though it could be defeated by experts; computers did not defeat champions until the late 1980s. While 10-digit phone numbers for long-distance national dialing originated in 1951, longer phone numbers for international dialing became a reality in 1970. Installation of personal in-flight entertainment displays by major airlines began in the early-to-mid 1990s, offering video games, TV broadcasts and movies in a manner similar to that shown in the film. The film also shows flat-screen TV monitors, of which the first real-world prototype appeared in 1972 produced by Westinghouse, but was not used for broadcast television until 1998. Plane cockpit integrated system displays, known as "glass cockpits", were introduced in the 1970s |
1304_24 | (originally in NASA Langley's Boeing 737 Flying Laboratory). Today such cockpits appear not only in high-tech aircraft like the Boeing 777, but have also been employed in space shuttles, the first being Atlantis in 1985. Rudimentary voice-controlled computing began in the early 1980s with the SoftVoice Computer System and existed in more sophisticated form by the early 2000s, although not as sophisticated as depicted in the film. The first picture phone was demonstrated at the 1964 New York World's Fair; however, due to the bandwidth limitations of telephone lines, personal video communication did not succeed commercially and has only been practical over broadband internet connections. Personal (audio) wireless telephones were ubiquitous in 2001, and yet no one in the movie had a small personal communication device. |
1304_25 | Some technologies portrayed as common in the film which had not materialized in the 2000s include commonplace civilian space travel, space stations with hotels, Moon colonization, suspended animation of humans, practical nuclear propulsion in spacecraft and strong artificial intelligence of the kind displayed by Hal.
Companies and countries
There are corporate logos and entities in the film that either didn't exist, no longer exist, or were broken up by anti-trust lawsuits. Still others changed their business model or represent countries that no longer exist. |
1304_26 | The British Broadcasting Corporation never expanded to have a BBC-12. BBC Three and Four came into existence in 2003 and 2002 respectively and newer channels used names such as BBC News and BBC Parliament. The corporations IBM, Aeroflot, Howard Johnson's, Whirlpool Corporation and Hilton Hotels, visual references of which appear in the film, have survived beyond 2001, although by 2001 Howard Johnson's had switched its business focus to hotels, rather than the restaurants shown in the film. The film depicts a still-existing Pan Am (which went out of business in 1991) and a still-existing Bell System telephone company (which was broken up in 1984 as a result of an anti-monopoly lawsuit filed by the U.S. Justice Department). The Bell System logo seen in the film was modified in 1969 and dropped entirely in 1983.
See also
Technology in science fiction
References
External links |
1304_27 | 2001: A Space Odyssey Internet Resource Archive
The 2001: A Space Odyssey Collectibles Exhibit
The Alt.Movies.Kubrick FAQ many observations on the meaning of 2001
The Kubrick Site including many works on 2001
American Institute of Aeronautics, 40 Anniversary article in Houston Section, Horizons, April 2008
Fictional technology by work
Space Odyssey |
1305_0 | Chocoladefabriken Lindt & Sprüngli AG, more commonly known simply as Lindt, is a Swiss chocolatier and confectionery company founded in 1845 and known for its chocolate truffles and chocolate bars, among other sweets. It is based in Kilchberg, where its main factory and museum are located.
History
Founding and early years
The origins of the company date back to 1836, when David Sprüngli-Schwarz and his son Rudolf Sprüngli-Ammann bought a small confectionery shop in the old town of Zürich, producing chocolates under the name David Sprüngli & Son. Before they moved to Paradeplatz in 1845, they established a small factory where they produced their chocolate in solidified form in 1838. |
1305_1 | When Rudolf Sprüngli-Ammann retired in 1892, he gave two equal parts of the business to his sons. The younger brother David Robert received two confectionery stores that became known under the name Confiserie Sprüngli. The elder brother Johann Rudolf received the chocolate factory. To raise the necessary finances for his expansion plans, Johann Rudolf then converted his private company into "Chocolat Sprüngli AG" in 1899. In that same year, he acquired the chocolate factory of Rodolphe Lindt in Bern, and the company changed its name to "Aktiengesellschaft Vereinigte Berner und Zürcher Chocoladefabriken Lindt & Sprüngli" (United Bern and Zurich Lindt and Sprungli Chocolate Factory Ltd.). |
1305_2 | Expansion
In 1994, Lindt & Sprüngli acquired the Austrian chocolatier Hofbauer Österreich and integrated it, along with its Küfferle brand, into the company. In 1997 and 1998, respectively, the company acquired the Italian chocolatier Caffarel and the American chocolatier Ghirardelli, and integrated both of them into the company as wholly-owned subsidiaries. Since then, Lindt & Sprüngli has expanded the once-regional Ghirardelli to the international market.
On 17 March 2009, Lindt announced the closure of 50 of its 80 retail boutiques in the United States because of weaker demand in the wake of the late-2000s recession.
Recent developments
On 14 July 2014, Lindt bought Russell Stover Candies, maker of Whitman's Chocolate, for about $1 billion, the company's largest acquisition to date.
In November 2018, Lindt opened its first American travel retail store in JFK Airport's Terminal 1 and its flagship Canadian shop in Yorkdale Shopping Centre, Toronto. |
1305_3 | Factories
Lindt & Sprüngli has 12 factories: Kilchberg, Switzerland; Aachen, Germany; Oloron-Sainte-Marie, France; Induno Olona, Italy; Gloggnitz, Austria; and Stratham, New Hampshire, in the United States. The factory in Gloggnitz, Austria, manufactures products under the Hofbauer & Küfferle brand in addition to the Lindt brand. Caffarel's factory is located in Luserna San Giovanni, Italy, and Ghirardelli's factory is located in San Leandro, California, in the United States. Furthermore, there are four more factories of Russell Stover in the United States including locations in Corsicana, Texas, Abilene, Kansas, and Iola, Kansas.
Since 2020, the main factory of Kilchberg includes a visitor centre and museum, referred to as Lindt Home of Chocolate. The museum notably displays the world's largest chocolate fountain, measuring over nine metres tall and containing 1,500 litres of chocolate, flowing from a giant whisk.
Lindt chocolate cafés |
1305_4 | Lindt has opened over 410 chocolate cafés and shops all over the world. The cafés' menu mostly focuses on chocolate and desserts. Lindt chocolate cafés also sell handmade chocolates, macaroons, cakes, and ice cream.
On 15 December 2014, 18 people, including eight staff, were held hostage at a Lindt cafe in Sydney. Three people, including the gunman, died in the incident.
Products
Lindor
Lindor is a Lindt's brand introduced as a chocolate bar in 1949 and later in 1967 in the form of a chocolate truffle. It is now characterized by a hard chocolate shell and a smooth chocolate filling. The ball or bar are available in an array of flavors, each with a distinctive color wrapper:
Most of the US Lindor truffles are manufactured in Stratham, New Hampshire.
In 2009, Swiss tennis star Roger Federer was named as Lindt's "global brand ambassador", and began appearing in a series of commercials endorsing Lindor.
Seasonal confectioneries |
1305_5 | The Lindt group also produces the Gold Bunny, a hollow milk chocolate rabbit in a variety of sizes available every Easter since 1952. Each bunny wears a small colored ribbon bow around its neck identifying the type of chocolate contained within. The milk chocolate bunny wears a red ribbon, the dark chocolate bunny wears a dark brown ribbon, the hazelnut bunny wears a green ribbon, and the white chocolate bunny wears a white ribbon. Other chocolates are wrapped to look like carrots, chicks, or lambs. The lambs are packaged with four white lambs and one black lamb.
During the Christmas season, Lindt produces a variety of items, including chocolate reindeer (which somewhat resemble the classic bunny), Santa, snowmen figures of various sizes, bears, bells, advent calendars, and chocolate ornaments. Various tins and boxes are available in the Lindt stores, the most popular colour schemes being the red and blue. Other seasonal items include Lindt chocolate novelty golf balls. |
1305_6 | For St. Valentine's Day, Lindt sells a boxed version of the Gold Bunny, which comes as a set of two kissing bunnies. Other Valentine's Day seasonal items include a selection of heart-shaped boxes of Lindt chocolate truffles.
Due to the 3rd UK lockdown and with all chocolate shops shut, Lindt has begun offering virtual tasting sessions (Feb 2021) with LINDT EXCELLENCE chocolatiers. The experiences can be bought online from their website and include a box of chocolates which are delivered, along with details of how to book the one-hour tasting.
Chocolate bars |
1305_7 | Lindt sells a variety of chocolate bars. Flavors from the Excellence range include:
Mint Intense: dark chocolate infused with mint
Lime Intense: dark chocolate infused with lime
Orange Intense: dark chocolate infused with orange essence and almond flakes
Blackcurrant: dark chocolate infused with pieces of blackcurrant and almond slivers
White Coconut: white chocolate with crisp flakes of fine coconut
Coconut: dark chocolate with crisp flakes of coconut
Almond: white chocolate with whole roasted almonds and caramelized almond pieces
Poire Intense: pear flavoured chocolate with almond flakes
Pineapple: dark chocolate with pineapple pieces and caramelized hazelnut pieces
Cherry Intense
Regular Dark Chocolate: available in 50%, 60%, 70%, 78%, 85%, 90%, 95%, 99%, or 100% cocoa varieties
Extra Creamy: milk chocolate
Toffee Crunch: crunchy toffee bits wrapped in milk chocolate
Caramel Crunchy: studded with crunchy caramel
Lindor: the famous balls but in cube form |
1305_8 | Wasabi: an East Asian-inspired dark chocolate mixed with wasabi
Pistachio: milk chocolate with creamy pistachio filling
Mandarin: milk chocolate with creamy mandarin filling
Strawberry: milk chocolate with creamy white chocolate strawberry filling
Strawberry Margarita: capsule form with strawberry and margarita filling
White Strawberry: white chocolate with strawberry pieces
Orange: milk chocolate with creamy orange-flavoured filling
Cuba: 55% cocoa, single-origin Cuban cocoa
Madagascar: 70% cocoa, single-origin Madagascar cocoa
Ecuador: 75% cocoa, single-origin Ecuadorian cocoa
Vanilla: white chocolate with vanilla beans
Coffee
Chili: 70% cocoa dark chocolate with red chilli extract
Raspberry Intense Dark: dark chocolate with pieces of raspberries and almond slivers
A Touch of Sea Salt: dark chocolate seasoned with fleur de sel |
1305_9 | Petits desserts
Lindt "Petits Desserts" range embodies famous European desserts in a small cube of chocolate. Flavors include: Tarte au Chocolat, Crème Brûlée, Tiramisu, Creme Caramel, Tarte Citron, Meringue, and Noir Orange.
Lindt makes a "Creation" range of chocolate-filled cubes: Milk Mousse, Dark Milk Mousse, White Milk Mousse, Chocolate Mousse, Orange Mousse, Pistachio and Cherry/Chili.
Liqueurs
Bâtons Kirsch are Lindt Kirsch liqueur-filled, chocolate-enclosed tubes dusted in cocoa powder.
Ice cream
In Australia, Lindt manufactures ice cream in various flavors:
70% Dark Chocolate
White Chocolate Framboise
Sable Cookies and Cream
Chocolate Chip Hazelnut
White Chocolate and Vanilla Bean
Products gallery |
1305_10 | Criticism
In September 2017, an investigation conducted by NGO Mighty Earth found that a large amount of the cocoa used in chocolate produced by Lindt and other major chocolate companies was grown illegally in national parks and other protected areas in the Ivory Coast and Ghana, the world's two largest cocoa producers. Mighty Earth's 2019 annual "Easter Chocolate Shopping Guide" awarded The Good Egg Award to Lindt "for greatest improvement in sustainable policies". |
1305_11 | In August 2020, the Federal Antimonopoly Service of Russia (FAS) opened up an antitrust case against Lindt after a failed response from the company a year earlier. The regulators have found quality differences for the same Lindt products in Russia over what is being sold in Western Markets without informing Russian consumers. According to the FAS, such behavior of foreign producers can lead to a redistribution of demand in the market and lead to unjustified benefits over other competitors, as companies like Lindt can still garner Russian demand for their products through brand recognition alone without delivering the same quality as in Western Europe. Lindt responded and denied that there are differences for its products sold in Russia and the EU, except for labeling. |
1305_12 | Share price
As of 2022 the share price, which is publicly traded in the United States over-the-counter market, is $103,100.00, it primarily trades on the SIX in Zurich. It is one of the most expensive stocks behind Berkshire Hathaway, but more expensive than Seaboard Corp, Amazon, NVR, and Google.
See also
List of bean-to-bar chocolate manufacturers
References
External links
Brand name chocolate
Swiss chocolate companies
Companies established in 1845
Companies listed on the SIX Swiss Exchange
Food and drink companies established in 1845
Luxury brands
Multinational companies headquartered in Switzerland
Swiss brands
Swiss confectionery
Companies based in the canton of Zürich
Museums in the canton of Zürich |
1306_0 | John William Davis (April 13, 1873 – March 24, 1955) was an American politician, diplomat and lawyer. He served under President Woodrow Wilson as the Solicitor General of the United States and the United States Ambassador to the United Kingdom. He was the Democratic nominee for president in 1924 but lost to Republican incumbent Calvin Coolidge. |
1306_1 | Born and raised in West Virginia, Davis briefly worked as a teacher before beginning his long legal career. Davis's father, John J. Davis, had been a delegate to the Wheeling Convention and served in Congress in the 1870s. Davis joined his father's legal practice and adopted many of his father's political views, including opposition to anti-lynching legislation and support for states' rights. Davis served in the United States House of Representatives from 1911 to 1913, helping to write the Clayton Antitrust Act. He held the position of solicitor general in the Justice Department from 1913 to 1918, during which time he successfully argued for the unconstitutionality of Oklahoma's "grandfather law" in Guinn v. United States, which had a discriminatory effect against African American voters. |
1306_2 | While serving as the ambassador to Britain from 1918 to 1921, Davis was a dark horse candidate for the 1920 Democratic presidential nomination. After he left office, Davis helped establish the Council on Foreign Relations and advocated for the repeal of Prohibition. The 1924 Democratic National Convention nominated Davis for president after 103 ballots. Davis remains the only major party presidential candidate from West Virginia. Running on a ticket with Charles W. Bryan, Davis lost in a landslide to incumbent President Coolidge. |
1306_3 | Davis did not seek public office again after 1924. He continued as a prominent attorney, representing many of the country's largest businesses. Over a 60-year legal career, he argued 140 cases before the United States Supreme Court. He notably argued the winning side in Youngstown Sheet & Tube Co. v. Sawyer, in which the Supreme Court ruled against President Harry Truman's seizure of the nation's steel plants. Davis unsuccessfully defended the "separate but equal" doctrine in Briggs v. Elliott, one of the companion cases to Brown v. Board of Education, in which the Court ruled in 1954 that segregated public schools were unconstitutional.
Family and early life |
1306_4 | Family background
Davis's paternal family had roots in western Virginia and what became West Virginia. His great-grandfather, Caleb Davis, was a clockmaker in the Shenandoah Valley. In 1816, his grandfather, John Davis, moved to Clarksburg in what would later become West Virginia. Its population then was 600–700, and he ran a saddle and harness business. His father, John James Davis, attended Lexington Law School, which later became the Washington and Lee University School of Law. By the age of twenty, he had established a law practice in Clarksburg. John J. Davis was a delegate in the Virginia General Assembly, and after the northwestern portion of Virginia broke away from the rest of Virginia in 1863 and formed West Virginia, he was elected to the new state's House of Delegates and later to the United States House of Representatives. |
1306_5 | John W. Davis's mother Anna Kennedy (1841–1917) was from Baltimore, Maryland, daughter of "William" Wilson Kennedy and his wife Catherine Esdale Martin. Kennedy was a lumber merchant. Catherine was the daughter of Tobias Martin, dairy farmer and amateur poet, and his wife, a member of the Esdale family. The Esdales were members of the Religious Society of Friends, or Quakers, who had settled near Valley Forge, Pennsylvania. They had reportedly helped provide support for the Continental Army under George Washington, which had camped there in the winter of 1777–1778.
Early years
Davis's Sunday school teacher recalled that "John W. Davis had a noble face even when small." His biographer said, "[h]e used better English, kept himself cleaner, and was more dignified than most youngsters. He was also extraordinarily well-mannered."
Education |
1306_6 | Davis's education began at home, as his mother taught him to read before he had memorized the alphabet. She had him read poetry and other literature from their home library. After turning ten, Davis was put in a class with older students to prepare him for the state teachers examination. A few years later, he was enrolled in a previously all-female seminary, that doubled as a private boarding and day school. He never had grades under 94.
Davis entered Washington and Lee University at the age of sixteen. He graduated in 1892 with a major in Latin. He joined the Phi Kappa Psi fraternity, participated in intramural sports, and attended mixed parties. |
1306_7 | He would have started law school directly after graduation, but he lacked funds. Instead, he became a teacher for Major Edward H. McDonald of Charles Town, West Virginia. Davis taught McDonald's nine children and his six nieces and nephews. His student Julia, nineteen at the time, later became Davis's wife. Davis fulfilled a nine-month contract with McDonald.
He returned to Clarksburg and apprenticed at his father's law practice. For fourteen months he copied documents by hand, read cases, and did much of what other aspiring lawyers did at the time to "read the law". |
1306_8 | Davis graduated with a law degree from Washington and Lee University School of Law in 1895 and was elected Law Class Orator. His speech gave a glimpse of his advocacy skills:
[The] lawyer has been always the sentinel of the watchtower of liberty. In all times and all countries has he stood forth in defense of his nation, her laws and liberties, not, it may be, under a shower of leaden death, but often with the frown of a revengeful and angry tyrant bent upon him.
Fellow classmates of 1895, shall we ... prove unworthy?
Early legal career
After graduation, Davis obtained the three signatures required to receive his law license (one from a local judge, and two from local attorneys, who attested to his proficiency in the law and upstanding moral character) and joined his father in practice in Clarksburg. They called their partnership Davis and Davis, Attorneys at Law. Davis lost his first three cases before his fortunes began to turn. |
1306_9 | Before Davis had completed his first year of private practice, he was recruited to Washington & Lee Law School as an assistant professor, starting in the fall of 1896. At the time, the law school had a faculty of two, and Davis became the third. At the end of the year, Davis was asked to return but demurred. He decided that he needed the "rough & tumble" of private practice.
Family connections
On June 20, 1899, he married Julia T. McDonald, who died on August 17, 1900. They had one daughter together, Julia McDonald Davis. She later married Charles P. Healy, and then William M. Adams. Several years later, the widower Davis married again, on January 2, 1912, to Ellen G. Bassel. She died in 1943.
Davis was the cousin and adoptive father of Cyrus Vance, who later served as Secretary of State under Jimmy Carter. |
1306_10 | Davis' daughter Julia was one of the first two female journalists hired by the Associated Press in 1926. (The other was probably Marguerite Young.) As noted above, Julia married William McMillan Adams, president of Sprague International the export subsidiary of Sprague Electric. He was the son of Arthur Henry Adams, president of the United States Rubber Company. Both father and son were aboard the luxury liner RMS Lusitania when it was sunk by a German submarine in 1915. Arthur died; his son William survived.
Julia and William divorced, and both remarried. She divorced again, and later they remarried in their old age. Adams had two sons by his second wife, John Perry and Arthur Henry Adams II. Julia died in 1993 with no natural children but claimed six "by theft and circumstance."
Political and diplomatic career |
1306_11 | Early career |
1306_12 | His father had been a delegate to the Wheeling Convention, which had created the state of West Virginia, but he had also opposed the abolitionists, Radical Republicans, and opposed ratification of the Fifteenth Amendment. Davis acquired much of his father's southern Democratic politics, opposing women's suffrage, Federal child-labor laws and anti-lynching legislation, Harry S. Truman's civil rights program, and defended the State's rights to establish the poll tax by questioning whether uneducated non-taxpayers should be allowed to vote. He was as much opposed to centralism in politics as he was to the concentration of capital by large corporations, supporting a number of early progressive laws regulating interstate commerce and limiting the power of corporations. Consequently, he felt distinctly out of place in the Republican Party, which supported free-association and free markets and maintained his father's staunch allegiance to the Democratic Party, even as he later represented |
1306_13 | the interests of business opposed to the New Deal. Davis ranked as one of the last Jeffersonians, as he supported states' rights and opposed a strong executive (he would be the lead attorney against Truman's nationalization of the steel industry). |
1306_14 | He represented West Virginia in the U.S. House of Representatives from 1911 to 1913, where he was one of the authors of the Clayton Antitrust Act. Davis also served as one of the managers in the successful impeachment trial of Judge Robert W. Archbald. He served as U.S. Solicitor General from 1913 to 1918. As Solicitor General, he successfully argued in Guinn v. United States for the illegality of Oklahoma's "grandfather law". That law exempted residents descended from a voter registered in 1866 (i.e. whites) from a literacy test which effectively disenfranchised blacks. Davis's personal posture differed from his position as an advocate. Throughout his career, he could separate his personal views and professional advocacy. |
1306_15 | Davis served as Wilson's ambassador to Great Britain from 1918 to 1921, he reflected deep Southern support for Wilsonianism, based on a reborn Southern patriotism, a distrust of the Republican Party, and a resurgence of Anglophilism. Davis proselytized in London for the League of Nations based on his paternalistic belief that peace depended primarily on Anglo-American friendship and leadership. He was disappointed by Wilson's mismanagement of the treaty ratification and by Republican isolationism and distrust of the League.
Presidential candidate |
1306_16 | Davis was a dark horse candidate for the Democratic nomination for President in both 1920 and 1924. His friend and partner Frank Polk managed his campaign at the 1924 Democratic National Convention. He won the nomination in 1924 as a compromise candidate on the one hundred and third ballot. Although Tennessee's Andrew Johnson served as President after Lincoln was assassinated, Davis' nomination made him the first presidential candidate from a former slave state since the Civil War, and as of 2020 he remains the only ever candidate from West Virginia. Davis' denunciation of the Ku Klux Klan and prior defense of black voting rights as Solicitor General under Wilson cost him votes in the South and among conservative Democrats elsewhere. He lost in a landslide to Calvin Coolidge, who did not leave the White House to campaign. Davis' 28.8 percent remains the smallest percentage of the popular vote ever won by a Democratic presidential nominee. He won every state of the former Confederacy |
1306_17 | and Oklahoma. |
1306_18 | Later political involvement
Davis was a member of the National Advisory Council of the Crusaders, an influential organization that promoted the repeal of prohibition. He was the founding President of the Council on Foreign Relations, formed in 1921, Chairman of the Carnegie Endowment for International Peace, and a trustee of the Rockefeller Foundation from 1922 to 1939. Davis also served as a delegate from New York to the 1928 and 1932 Democratic National Conventions.
Davis campaigned on behalf of Franklin D. Roosevelt in the 1932 presidential election but never developed a close relationship with Roosevelt. After Roosevelt took office, Davis quickly turned against the New Deal and joined with Al Smith and other anti-New Deal Democrats in forming the American Liberty League. He later supported the Republican presidential candidate in the 1936, 1940, and 1944 elections. |
1306_19 | Davis was implicated by retired Marine Corps Major General Smedley Butler in the Business Plot, an alleged political conspiracy in 1933 to overthrow United States President Franklin D. Roosevelt, in testimony before the McCormack-Dickstein Committee, whose deliberations began on November 20, 1934 and culminated in the Committee's report to the United States House of Representatives on February 15, 1935. Davis was not called before the committee because "The committee will not take cognizance of names brought into the testimony which constitute mere hearsay." |
1306_20 | In 1949, Davis (as a member of the board of the Carnegie Endowment for International Peace) testified as a character witness for Alger Hiss (Carnegie's president) during his trials (part of the Hiss-Chambers Case): "In the twilight of his career, following the end of World War II, Davis publicly supported Alger Hiss and J. Robert Oppenheimer during the hysteria of the McCarthy hearings" (more accurately, the "McCarthy Era" as the Hiss Case (1948–1950) preceded McCarthyism in the 1950s).
Legal career |
1306_21 | Davis was one of the most prominent and successful lawyers of the first half of the 20th century, arguing 140 cases before the U.S. Supreme Court. His firm, variously titled Stetson Jennings Russell & Davis, then Davis Polk Wardwell Gardiner & Reed, then Davis Polk Wardwell Sunderland & Kiendl (now Davis Polk & Wardwell), represented many of the largest companies in the United States in the 1920s and following decades. From 1931 to 1933, Davis also served as president of the New York City Bar Association.
In 1933, Davis served as legal counsel for the financier J.P. Morgan, Jr. and his companies during the Senate investigation into private banking and the causes of the recent Great Depression.
The last twenty years of Davis's practice included representing large corporations before the United States Supreme Court challenging the constitutionality and application of New Deal legislation. Davis lost many of these battles. |
1306_22 | Appearances before the U.S. Supreme Court
Davis argued 140 cases before the U.S. Supreme Court during his career. Seventy-three were as Solicitor General, and 67 as a private lawyer. Lawrence Wallace, who retired from the Office of the Solicitor General in 2003, argued 157 cases during his career but many believe that few attorneys have argued more cases than Davis. Daniel Webster and Walter Jones are believed to have argued more cases than Davis, but they were lawyers of a much earlier era.
Youngstown Steel case
One of Davis' most influential arguments before the Supreme Court was in Youngstown Sheet & Tube Co. v. Sawyer in May 1952, when the Court ruled on Truman's seizure of the nation's steel plants. |
1306_23 | While Davis wasn't brought into the case until March 1952, he was already familiar with the concept of a presidential seizure of a steel mill. In 1949, the Republic Steel Company, fearful of advice given to President Truman by Attorney General Tom C. Clark, asked Davis for an opinion letter on whether the President could seize private industry in a "National Emergency." Davis wrote that the President could not do so, unless such power already was vested in the President by law. He further went on to opine on the Selective Service Act of 1948's intent, and that seizures were only authorized if a company did not sufficiently prioritize government production in a time of crisis. |
1306_24 | Arguing for the steel industry, Davis spoke for eighty-seven minutes before the Court. He described Truman's acts as a usurpation' of power, that were 'without parallel in American history. The Justices allowed him to proceed uninterrupted, with only one question from Justice Frankfurter, who may have had a personal feeling against Davis relating to his 1924 presidential campaign. It had been predicted that the President's actions would be upheld, and the injunction would be lifted, but the Court decided 6–3, to uphold the injunction stopping the seizure of the steel mills.
Washington Post writer Chalmers Roberts subsequently wrote that rarely "has a courtroom sat in such silent admiration for a lawyer at the bar" in reference to Davis' oral argument. Unfortunately, Davis did not allow the oral argument to be printed because the stenographic transcript was so garbled he feared it would not be close to what was said at the Court. |
1306_25 | Of particular note in the case is that one of the Justices in the majority was Tom Clark, who as Attorney General in 1949 had advised Truman to proceed with the seizure of Republic Steel. Yet in 1952 Justice Clark voted with the majority without joining Black's opinion, in direct opposition to his previous advice.
Brown v. Board of Education |
1306_26 | Davis' legal career is most remembered for his final appearance before the Supreme Court, in which he unsuccessfully defended the "separate but equal" doctrine in Briggs v. Elliott, a companion case to Brown v. Board of Education. Davis, as a defender of racial segregation and state control of education, uncharacteristically displayed his emotions in arguing that South Carolina had shown good faith in attempting to eliminate any inequality between black and white schools and should be allowed to continue to do so without judicial intervention. He expected to win, most likely through a divided Supreme Court, even after the matter was re-argued after the death of Chief Justice Fred M. Vinson. After the Supreme Court unanimously ruled against his client's position, he returned the $25,000 (equivalent to $ in ), that he had received from South Carolina, although he was not required to do so, but kept a silver tea service that had been presented to him. It has also been reported that he |
1306_27 | never charged South Carolina in the first place. He declined to participate further in the case, as he did not wish to be involved in the drafting of decrees to implement the Court's decision. |
1306_28 | Death and legacy
Davis had been a member of the American Bar Association, the Council on Foreign Relations, Freemasons, Phi Beta Kappa, and Phi Kappa Psi. He was a resident of Nassau County, New York, and practiced law in New York City until his death in Charleston, South Carolina, at the age of 81. He is interred at Locust Valley Cemetery in Locust Valley, New York.
The John W. Davis Federal building on West Pike street in Clarksburg, West Virginia, is named after Davis.
The building housing the Student Health Center at Washington and Lee University is named for him, as is the Law School's appellate advocacy program, and an award for the graduating student with the highest grade point average
In the 1991 television film Separate but Equal, a dramatization of the Brown case, Davis was portrayed by the famed actor Burt Lancaster in his final film role. |
1306_29 | Electoral history
West Virginia's 1st congressional district, 1910:
John W. Davis (D) – 20,370 (48.9%)
Charles E. Carrigan (R) – 16,962 (40.7%)
A. L. Bauer (Socialist) – 3,239 (7.8%)
Ulysses A. Clayton (Prohibition) – 1,099 (2.6%)
West Virginia's 1st congressional district, 1912:
John W. Davis (D) (inc.) – 24,777 (45.0%)
George A. Laughlin (R) – 24,613 (44.7%)
D. M. S. Scott (Socialist) – 4,230 (7.7%)
L. E. Peters (Prohibition) – 1,482 (2.7%)
1924 Democratic presidential primaries
William McAdoo – 562,601 (56.1%)
Oscar W. Underwood – 77,583 (7.7%)
James M. Cox – 74,183 (7.4%)
Unpledged – 59,217 (5.9%)
Henry Ford – 49,737 (5.0%)
Thomas J. Walsh – 43,108 (4.3%)
Woodbridge Nathan Ferris – 42,028 (4.2%)
George Silzer – 35,601 (3.6%)
Al Smith – 16,459 (1.6%)
L. B. Musgrove – 12,110 (1.2%)
William Dever – 1,574 (0.2%)
James A. Reed – 84 (0.0%)
John W. Davis – 21 (0.0%) |
1306_30 | 1924 United States presidential election
Calvin Coolidge/Charles G. Dawes (R) – 15,723,789 (54.0%) and 382 electoral votes (35 states carried)
John W. Davis/Charles W. Bryan (D) – 8,386,242 (28.8%) and 136 electoral votes (12 states carried)
Robert M. La Follette, Sr./Burton K. Wheeler (Progressive) – 4,831,706 (16.6%) and 13 electoral votes (1 state carried)
See also
Brown v. Board of Education
Guinn v. United States
Youngstown Sheet & Tube Co. v. Sawyer
References
Further reading
External links
John W. Davis papers (MS 170). Manuscripts and Archives, Yale University Library.
International Home of the English-Speaking Uni
CFR Website – Continuing the Inquiry: The Council on Foreign Relations from 1921 to 1996 The history of the Council by Peter Grose, a Council member.
Website of Davis Polk & Wardwell, law firm of which Davis was a member and which bears his name today
Political Graveyard
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1306_31 | 1873 births
1955 deaths
20th-century American politicians
Ambassadors of the United States to the United Kingdom
Burials at Locust Valley Cemetery
Davis Polk & Wardwell lawyers
Democratic Party (United States) presidential nominees
Democratic Party members of the United States House of Representatives
Honorary Knights Commander of the Order of the British Empire
Members of the Council on Foreign Relations
Members of the United States House of Representatives from West Virginia
Old Right (United States)
Politicians from Clarksburg, West Virginia
Presidents of the Council on Foreign Relations
Presidents of the New York City Bar Association
Candidates in the 1920 United States presidential election
Candidates in the 1924 United States presidential election
United States Solicitors General
Washington and Lee University School of Law alumni
Washington and Lee University School of Law faculty
West Virginia Democrats
Woodrow Wilson administration personnel
American white supremacists |
1306_32 | 20th-century American diplomats |
1307_0 | In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value.
As a consequence, it allows the order of integration to be changed in certain iterated integrals.
Fubini's theorem implies that two iterated integrals are equal to the corresponding double integral across its integrands. Tonelli's theorem, introduced by Leonida Tonelli in 1909, is similar, but applies to a non-negative measurable function rather than one integrable over their domains.
A related theorem is often called Fubini's theorem for infinite series, which states that if is a doubly-indexed sequence of real numbers, and if is absolutely convergent, then |
1307_1 | Although Fubini's theorem for infinite series is a special case of the more general Fubini's theorem, it is not appropriate to characterize it as a logical consequence of Fubini's theorem. This is because some properties of measures, in particular sub-additivity, are often proved using Fubini's theorem for infinite series. In this case, Fubini's general theorem is a logical consequence of Fubini's theorem for infinite series.
History
The special case of Fubini's theorem for continuous functions on a product of closed bounded subsets of real vector spaces was known to Leonhard Euler in the 18th century. extended this to bounded measurable functions on a product of intervals. conjectured that the theorem could be extended to functions that were integrable rather than bounded, and this was proved by . gave a variation of Fubini's theorem that applies to non-negative functions rather than integrable functions. |
1307_2 | Product measures
If X and Y are measure spaces with measures, there are several natural ways to define a product measure on their product.
The product X × Y of measure spaces (in the sense of category theory) has as its measurable sets the σ-algebra generated by the products A × B of measurable subsets of X and Y. |
1307_3 | A measure μ on X × Y is called a product measure if μ(A × B) = μ1(A)μ2(B) for measurable subsets A ⊂ X and B ⊂ Y and measures µ1 on X and µ2 on Y. In general there may be many different product measures on X × Y. Fubini's theorem and Tonelli's theorem both need technical conditions to avoid this complication; the most common way is to assume all measure spaces are σ-finite, in which case there is a unique product measure on X×Y. There is always a unique maximal product measure on X × Y, where the measure of a measurable set is the inf of the measures of sets containing it that are countable unions of products of measurable sets. The maximal product measure can be constructed by applying Carathéodory's extension theorem to the additive function μ such that μ(A × B) = μ1(A)μ2(B) on the ring of sets generated by products of measurable sets. (Carathéodory's extension theorem gives a measure on a measure space that in general contains more measurable sets than the measure space X × Y, so |
1307_4 | strictly speaking the measure should be restricted to the σ-algebra generated by the products A × B of measurable subsets of X and Y.) |
1307_5 | The product of two complete measure spaces is not usually complete. For example, the product of the Lebesgue measure on the unit interval I with itself is not the Lebesgue measure on the square I × I. There is a variation of Fubini's theorem for complete measures, which uses the completion of the product of measures rather than the uncompleted product.
For integrable functions
Suppose X and Y are σ-finite measure spaces, and suppose that X × Y is given the product measure (which is unique as X and Y are σ-finite). Fubini's theorem states that if f is X × Y integrable, meaning that f is a measurable function and
then
The first two integrals are iterated integrals with respect to two measures, respectively, and the third is an integral with respect to the product measure. The partial integrals and need not be defined everywhere, but this does not matter as the points where they are not defined form a set of measure 0. |
1307_6 | If the above integral of the absolute value is not finite, then the two iterated integrals may have different values. See below for an illustration of this possibility. |
1307_7 | The condition that X and Y are σ-finite is usually harmless because in practice almost all measure spaces one wishes to use Fubini's theorem for are σ-finite.
Fubini's theorem has some rather technical extensions to the case when X and Y are not assumed to be σ-finite . The main extra complication in this case is that there may be more than one product measure on X×Y. Fubini's theorem continues to hold for the maximal product measure, but can fail for other product measures. For example, there is a product measure and a non-negative measurable function f for which the double integral of |f| is zero but the two iterated integrals have different values; see the section on counterexamples below for an example of this. Tonelli's theorem and the Fubini–Tonelli theorem (stated below) can fail on non σ-finite spaces even for the maximal product measure. |
1307_8 | Tonelli's theorem for non-negative measurable functions
Tonelli's theorem (named after Leonida Tonelli) is a successor of Fubini's theorem. The conclusion of Tonelli's theorem is identical to that of Fubini's theorem, but the assumption that has a finite integral is replaced by the assumption that is a non-negative measurable function.
Tonelli's theorem states that if (X, A, μ) and (Y, B, ν) are σ-finite measure spaces, while f from X×Y to [0,∞] is non-negative measurable function, then
A special case of Tonelli's theorem is in the interchange of the summations, as in , where are non-negative for all x and y. The crux of the theorem is that the interchange of order of summation holds even if the series diverges. In effect, the only way a change in order of summation can change the sum is when there exist some subsequences that diverge to and others diverging to . With all elements non-negative, this does not happen in the stated example. |
1307_9 | Without the condition that the measure spaces are σ-finite it is possible for all three of these integrals to have different values. |
1307_10 | Some authors give generalizations of Tonelli's theorem to some measure spaces that are not σ-finite but these generalizations often add conditions that immediately reduce the problem to the σ-finite case. For example, one could take the σ-algebra on A×B to be that generated by the product of subsets of finite measure, rather than that generated by all products of measurable subsets, though this has the undesirable consequence that the projections from the product to its factors A and B are not measurable. Another way is to add the condition that the support of f is contained in a countable union of products of sets of finite measure. gives some rather technical extensions of Tonelli's theorem to some non σ-finite spaces. None of these generalizations have found any significant applications outside abstract measure theory, largely because almost all measure spaces of practical interest are σ-finite. |
1307_11 | Fubini–Tonelli theorem
Combining Fubini's theorem with Tonelli's theorem gives
the Fubini–Tonelli theorem (often just called Fubini's theorem), which states that if X and Y are σ-finite measure spaces, and if f is a measurable function, then
Besides if any one of these integrals is finite, then
The absolute value of f in the conditions above can be replaced by either the positive or the negative part of f; these forms include Tonelli's theorem as a special case as the negative part of a non-negative function is zero and so has finite integral. Informally all these conditions say that the double integral of f is well defined, though possibly infinite.
The advantage of the Fubini–Tonelli over Fubini's theorem is that the repeated integrals of the absolute value of |f| may be easier to study than the double integral. As in Fubini's theorem, the single integrals may fail to be defined on a measure 0 set. |
1307_12 | For complete measures
The versions of Fubini's and Tonelli's theorems above do not apply to integration on the product of the real line R with itself with Lebesgue measure. The problem is that Lebesgue measure on R×R is not the product of Lebesgue measure on R with itself, but rather the completion of this: a product of two complete measure spaces X and Y is not in general complete. For this reason one sometimes uses versions of Fubini's theorem for complete measures: roughly speaking one just replaces all measures by their completions. The various versions of Fubini's theorem are similar to the versions above, with the following minor differences:
Instead of taking a product X×Y of two measure spaces, one takes the completion of some product. |
1307_13 | If f is a measurable on the completion of X×Y then its restrictions to vertical or horizontal lines may be non-measurable for a measure zero subset of lines, so one has to allow for the possibility that the vertical or horizontal integrals are undefined on a set of measure 0 because they involve integrating non-measurable functions. This makes little difference, because they can already be undefined due to the functions not being integrable.
One generally also assumes that the measures on X and Y are complete, otherwise the two partial integrals along vertical or horizontal lines may be well-defined but not measurable. For example, if f is the characteristic function of a product of a measurable set and a non-measurable set contained in a measure 0 set then its single integral is well defined everywhere but non-measurable. |
1307_14 | Proofs |
1307_15 | Proofs of the Fubini and Tonelli theorems are necessarily somewhat technical, as they have to use a hypothesis related to σ-finiteness. Most proofs involve building up to the full theorems by proving them for increasingly complicated functions with the steps as follows.
Use the fact that the measure on the product is a product measure to prove the theorems for the characteristic functions of rectangles.
Use the condition that the spaces are σ-finite (or some related condition) to prove the theorem for the characteristic functions of measurable sets. This also covers the case of simple measurable functions (measurable functions taking only a finite number of values).
Use the condition that the functions are measurable to prove the theorems for positive measurable functions by approximating them by simple measurable functions. This proves Tonelli's theorem. |
1307_16 | Use the condition that the functions are integrable to write them as the difference of two positive integrable functions, and apply Tonelli's theorem to each of these. This proves Fubini's theorem. |
1307_17 | Riemann integrals
For Riemann integrals, Fubini's theorem is proven by refining the partitions along the x-axis and y-axis as to create a joint partition of the form , which is a partition over . This is used to show that the double integrals of either order are equal to the integral over .
Counterexamples
The following examples show how Fubini's theorem and Tonelli's theorem can fail if any of their hypotheses are omitted.
Failure of Tonelli's theorem for non σ-finite spaces |
1307_18 | Suppose that X is the unit interval with the Lebesgue measurable sets and Lebesgue measure, and Y is the unit interval with all subsets measurable and the counting measure, so that Y is not σ-finite. If f is the characteristic function of the diagonal of X×Y, then integrating f along X gives the 0 function on Y, but integrating f along Y gives the function 1 on X. So the two iterated integrals are different. This shows that Tonelli's theorem can fail for spaces that are not σ-finite no matter what product measure is chosen. The measures are both decomposable, showing that Tonelli's theorem fails for decomposable measures (which are slightly more general than σ-finite measures).
Failure of Fubini's theorem for non-maximal product measures |
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