In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length and volume. A important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space Rn. For instance, the Lebesgue measure of the interval in the real numbers is its length in the everyday sense of the word 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to subsets of a set X, it must further be countably additive: the measure of a'large' subset that can be decomposed into a finite number of'smaller' disjoint subsets is equal to the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure.
This problem was resolved by defining measure only on a sub-collection of all subsets. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined are complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, Maurice Fréchet, among others; the main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space. Probability theory considers measures that assign to the whole set the size 1, considers measurable subsets to be events whose probability is given by the measure.
Ergodic theory considers measures that are invariant under, or arise from, a dynamical system. Let X be a set and Σ a σ-algebra over X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties: Non-negativity: For all E in Σ: μ ≥ 0. Null empty set: μ = 0. Countable additivity: For all countable collections i = 1 ∞ of pairwise disjoint sets in Σ: μ = ∑ k = 1 ∞ μ One may require that at least one set E has finite measure; the empty set automatically has measure zero because of countable additivity, because μ = μ = μ + μ + μ + …, which implies that μ = 0. If only the second and third conditions of the definition of measure above are met, μ takes on at most one of the values ±∞ μ is called a signed measure; the pair is called a measurable space, the members of Σ are called measurable sets. If and are two measurable spaces a function f: X → Y is called measurable if for every Y-measurable set B ∈ Σ Y, the inverse image is X-measurable – i.e.: f ∈ Σ X.
In this setup, the composition of measurable functions is measurable, making the measurable spaces and measurable functions a category, with the measurable spaces as objects and the set of measurable functions as arrows. See Measurable function#Term usage variations about another setup. A triple is called a measure space. A probability measure is a measure with total measure one – i.e. Μ = 1. A probability space is a measure space with a probability measure. For measure spaces that are topological spaces various compatibility conditions can be
Patrick Paul Billingsley was an American mathematician and stage and screen actor, noted for his books in advanced probability theory and statistics. He was born and raised in Sioux Falls, South Dakota, graduated from the United States Naval Academy in 1946. After earning a Ph. D. in mathematics at Princeton University in 1955, he was attached to the NSA until his discharge from the Navy in 1957. In 1958 he became a professor of mathematics and statistics at the University of Chicago, where he served as chair of the Department of Statistics from 1980 to 1983, retired in 1994. In 1964–65 he was a Fulbright Fellow and visiting professor at the University of Copenhagen. In 1971 -- 72 he was visiting professor at the University of Cambridge. From 1976 to 1979 he edited the Annals of Probability. In 1983 he was president of the Institute of Mathematical Statistics, he was given the Lester R. Ford Award for his article "Prime Numbers and Brownian Motion." He was elected a Fellow of the American Academy of Arts and Sciences in 1986.
He starred in a number of plays at Court Theatre and Body Politic Theatre in Chicago and appeared in at least nine films. In Young Men and Fire, fellow University of Chicago professor Norman Maclean wrote about Billingsley that "he is a distinguished statistician and one of the best amateur actors I have seen". Statistical Inference for Markov Processes Ergodic Theory and Information Convergence of Probability Measures The Elements of Statistical Inference Probability and Measure Three Magic Keys, Taliesin The Pirates of Penzance, Pirate Read Me a Story Clue of the Circus Clowns, Circus Master Finian's Rainbow, Buzz Collins Beadle-Levi Show Guys and Dolls, Arvide Abernathy We Bombed in New Haven Victorian Children Vaudeville Show, singer The Threepenny Opera, street singer Four Plays of Fantasy and the Unusual Moulin Rouge Oh, What a Lovely War! Midsummer Night's Dream, Theseus The Caretaker, Aston The Father, The Captain Murder in the Cathedral, First Knight Twelfth Night, Feste The Same Room, Tom Ferris Dracula, Dr. Seward Much Ado about Nothing and Friar Frances Exits and Entrances Trifles, Sheriff Peters The Lover, Richard-Max The Tempest, Alonzo She Stoops to Conquer, Mr. Hardcastle Measure for Measure, The Duke Mrs. Warren's Profession, Rev. Samuel Gardiner Equus, Dr. Dysart The Seagull, Sorin Twelfth Night, Anotnio Under Milk Wood, Rev.
Eli Jenkins et al. The First Night of Pygmalion, Beerbohm-Tree et al. Midsummer Night's Dream, Peter Quince Much Ado about Nothing, Leonato Heartbreak House, Mazzini Dunn Every Good Boy Deserves Favor, KGB colonel The Birthday Party, Petey Arms and the Man, Major Petkoff Moonlight Daring Us to Go Insane, J. Earl Sheets Coastal Disturbances, R. Hamilton Adams The Fury - CIA agent Lander My Bodyguard - Biology Teacher Somewhere in Time - Professor Another Saturday Night - Mr. McGrath The Untouchables - Bailiff #2 Dummy - Dr. Morris Flesh and Blood - Boxing official The Children Nobody Wanted - Preacher The Dollmaker - Cooper The Last Leaf – A Parable of Easter - Dr. Winchester The Private Eye - Guard Murder Ordained - Ray Call Jack and Mike - Judge Sable - Sullivan The Father Clements Story - Father Donovan He died in 2011, aged 85, in his Hyde Park, Chicago home, he was survived by his children, Patty, Julie and Paul, his companion, Florence Weisblatt. His wife of nearly 50 years, social activist Ruth Billingsley, died in 2000.
Obituary in the Chicago Tribune Patrick Billingsley, probability theorist and actor, 1925–2011
John Wiley & Sons, Inc. branded as Wiley in recent years, is a global publishing company that specializes in academic publishing and instructional materials. The company produces books and encyclopedias, in print and electronically, as well as online products and services, training materials, educational materials for undergraduate and continuing education students. Founded in 1807, Wiley is known for publishing the For Dummies book series. In 2017, the company had a revenue of $1.7 billion. Wiley was established in 1807; the company was the publisher of such 19th century American literary figures as James Fenimore Cooper, Washington Irving, Herman Melville, Edgar Allan Poe, as well as of legal and other non-fiction titles. Wiley worked in partnership with Cornelius Van Winkle, George Long, George Palmer Putnam, Robert Halsted; the firm took its current name in 1865. Wiley shifted its focus to scientific and engineering subject areas, abandoning its literary interests. Charles Wiley's son John took over the business when his father died in 1826.
The firm was successively named Wiley, Lane & Co. Wiley & Putnam, John Wiley; the company acquired its present name in 1876, when John's second son William H. Wiley joined his brother Charles in the business. Through the 20th century, the company expanded its publishing activities, the sciences, higher education. Since the establishment of the Nobel Prize in 1901, Wiley and its acquired companies have published the works of more than 450 Nobel Laureates, in every category in which the prize is awarded. One of the world's oldest independent publishing companies, Wiley marked its bicentennial in 2007 with a year-long celebration, hosting festivities that spanned four continents and ten countries and included such highlights as ringing the closing bell at the New York Stock Exchange on May 1. In conjunction with the anniversary, the company published Knowledge for Generations: Wiley and the Global Publishing Industry, 1807-2007, depicting Wiley's pivotal role in the evolution of publishing against a social and economic backdrop.
Wiley has created an online community called Wiley Living History, offering excerpts from Knowledge for Generations and a forum for visitors and Wiley employees to post their comments and anecdotes. In December 2010, Wiley opened an office in Dubai; the company has had an office in Beijing, since 2001, China is now its sixth-largest market for STEM content. Wiley established publishing operations in India in 2006, has established a presence in North Africa through sales contracts with academic institutions in Tunisia and Egypt. On April 16, 2012, the company announced the establishment of Wiley Brasil Editora LTDA in São Paulo, effective May 1, 2012. Wiley's scientific and medical business was expanded by the acquisition of Blackwell Publishing in February 2007; the combined business, named Scientific, Technical and Scholarly, publishes, in print and online, 1,400 scholarly peer-reviewed journals and an extensive collection of books, major reference works and laboratory manuals in the life and physical sciences and allied health, the humanities, the social sciences.
Through a backfile initiative completed in 2007, 8.2 million pages of journal content have been made available online, a collection dating back to 1799. Wiley-Blackwell publishes on behalf of about 700 professional and scholarly societies. Other major journals published include Angewandte Chemie, Advanced Materials, International Finance and Liver Transplantation. Launched commercially in 1999, Wiley InterScience provided online access to Wiley journals, major reference works, books, including backfile content. Journals from Blackwell Publishing were available online from Blackwell Synergy until they were integrated into Wiley InterScience on June 30, 2008. In December 2007, Wiley began distributing its technical titles through the Safari Books Online e-reference service. On February 17, 2012, Wiley announced the acquisition of Inscape Holdings Inc. which provides DISC assessments and training for interpersonal business skills. Wiley described the acquisition as complementary to the workplace learning products published under its Pfeiffer imprint, one that would help Wiley advance its digital delivery strategy and extend its global reach through Inscape's international distributor network.
On March 7, 2012, Wiley announced its intention to divest assets in the areas of travel, general interest, nautical and crafts, as well as the Webster's New World and CliffsNotes brands. The planned divestiture was aligned with Wiley's "increased strategic focus on content and services for research and professional practices, on lifelong learning through digital technology". On August 13, 2012, Wiley announced it entered into a definitive agreement to sell all of its travel assets, including all of its interests in the Frommer's brand, to Google Inc. On November 6, 2012, Houghton Mifflin Harcourt acquired Wiley's cookbooks and study guides. In 2013, Wiley sold its pets and general interest lines to Turner Publishing Company and its nautical line to Fernhurst Books. H
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Harald Cramér was a Swedish mathematician and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of statistical theory". Harald Cramér was born in Stockholm, Sweden on 25 September 1893. Cramér remained close to Stockholm for most of his life, he entered the University of Stockholm as an undergraduate in 1912, where he studied mathematics and chemistry. During this period, he was a research assistant under the famous chemist, Hans von Euler-Chelpin, with whom he published his first five articles from 1913 to 1914. Following his lab experience, he began to focus on mathematics, he began his work on his doctoral studies in mathematics which were supervised by Marcel Riesz at the University of Stockholm. Influenced by G. H. Hardy, Cramér's research led to a PhD in 1917 for his thesis "On a class of Dirichlet series". Following his PhD, he served as an Assistant Professor of Mathematics at Stockholm University from 1917 to 1929.
Early on, Cramér was involved in analytic number theory. He made some important statistical contributions to the distribution of primes and twin primes, his most famous paper on this subject is entitled "On the order of magnitude of the difference between consecutive prime numbers", which provided a rigorous account of the constructive role in which probability applied to number theory and included an estimate for prime gaps that became known as Cramér's conjecture. In the late 1920s, Cramér became interested in the field of probability, which at the time was not an accepted branch of mathematics. Cramér knew that a radical change was needed in this field, in a paper in 1926 said, "The probability concept should be introduced by a purely mathematical definition, from which its fundamental properties and the classical theorems are deduced by purely mathematical operations." Cramér took an interest in the rigorous mathematical formulation of probability in the work of French and Russian mathematicians such as Kolmogorov, Lévy, Khinchin in the early 1930s.
Cramér made significant development to the revolution in probability theory. Cramér wrote his careful study of the field in his Cambridge publication Random variables and probability distributions which appeared in 1937. Shortly after World War II, Cramér went on to publish the influential Mathematical Methods of Statistics in 1946; this text was one that "showed the way in which statistical practice depended on a body of rigorous mathematical analysis as well as Fisherian intuition."In 1929, Cramér was appointed to a newly created chair in Stockholm University, becoming the first Swedish professor of Actuarial Mathematics and Mathematical Statistics. Cramér retained this position up until 1958. During his tenure at Stockholm University, Cramér was a PhD advisor for 10 students, most notably Herman Wold and Kai Lai Chung. In 1950 he was elected as a Fellow of the American Statistical Association. Starting in 1950, Cramér took on the additional responsibility of becoming the President of Stockholm University.
In 1958, he was appointed to be Chancellor of the entire Swedish university system. Cramér retired from the Swedish university system in 1961. A large portion of Cramér's work concerned the field of actuarial insurance mathematics. During the period from 1920 to 1929, he was an actuary for the life insurance company Svenska livförsäkringsbolaget, his actuarial work during this time led him to study probability and statistics which became the main area of his research. In 1927 he published some of its applications. Following his work for Svenska livförsäkringsbolaget, he went on to work for Återförsäkringsaktiebolaget Sverige, a reinsurance company, up until 1948, he was known for his pioneering efforts in insurance risk theory. After this period, he remained as a consultant actuary to Sverige from 1949 to 1961. In his life, he was elected to be the Honorary President of the Swedish Actuarial Society. Cramér remained an active contributor to his professional career for an additional 20 years. Following his retirement in 1961, he became active in research, slowed due to his Chancellorship.
During the years from 1961 to 1983, Cramér traveled throughout the United States and Europe to continue his research, making significant stops at Berkeley, at the Research Triangle Institute of North Carolina. Cramér received an Honorary Doctorate from Heriot-Watt University in 1972, his academic career spanned over seven decades, from 1913 to 1982. Harald Cramér married Marta Hansson in 1918, they remained together up until her death in 1973, he had referred to her as his "Beloved Marta". Together they had one daughter, Marie-Louise, two sons and Kim. Cramér, Harald. "Über eine Eigenschaft der normalen Verteilungsfunktion". Mathematische Zeitschrift. 41: 405–414. Doi:10.1007/BF01180430. MR1545629 Cramér, Harald. "Sur un nouveau théorème-limite de la théorie des probabilités". Actualités Scientifiques et Industrielles. 736: 5–23. Wegman, Edward. "Some Personal Recollections of Harald Cramér on the Development of Statistics and Probability". Statistical Science. 1: 528–535. Doi:10.1214/ss/1177013531. JSTOR 2245807.
Kingman, J. F. C.. "Harald Cramér, 1893-1985". Journal of the Royal Statistical Society. 149: 186. JSTOR 2981530. Blom, Gunnar. "Harald Cramér, 1893-1985". The Annals of Statistics. 15: 1335–1350. Doi:10.1214/aos/1176350596. JSTOR 2241677. Kendall, David. "A Tribute to Harald Cramér". Journal of the Royal Statistical Society, Series A. 146: 211–212. JSTOR 2981652. Heyde, C
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, measure, infinite series, analytic functions. These theories are studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary techniques of analysis. Analysis may be distinguished from geometry. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids; the explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century.
In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would be called Cavalieri's principle to find the volume of a sphere in the 5th century; the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine and arctangent. Alongside his development of the Taylor series of the trigonometric functions, he estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series, his followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century. The modern foundations of mathematical analysis were established in 17th century Europe. Descartes and Fermat independently developed analytic geometry, a few decades Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations and partial differential equations, Fourier analysis, generating functions.
During this period, calculus techniques were applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra used in earlier work by Euler. Instead, Cauchy formulated calculus in terms of geometric infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y, he introduced the concept of the Cauchy sequence, started the formal theory of complex analysis. Poisson, Liouville and others studied partial differential equations and harmonic analysis; the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis.
In the middle of the 19th century Riemann introduced his theory of integration. The last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, introduced the "epsilon-delta" definition of limit. Mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions. "monsters" began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, Baire proved the Baire category theorem.
In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, Hilbert introduced Hilbert spaces to solve integral equations; the idea of normed vector space was in the air, in the 1920s Banach created functional analysis. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. Much of analysis happens in some metric space. Examples of analysis without a metric include functional analysis. Formally, a metric space is an ordered pair where M is a set
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. The difference between a probability measure and the more general notion of measure is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events, e.g. the value assigned to "1 or 2" in a throw of a die should be the sum of the values assigned to "1" and "2". Probability measures have applications in diverse fields, from physics to biology; the requirements for a function μ to be a probability measure on a probability space are that: μ must return results in the unit interval, returning 0 for the empty set and 1 for the entire space.μ must satisfy the countable additivity property that for all countable collections of pairwise disjoint sets: μ = ∑ i ∈ I μ.
For example, given three elements 1, 2 and 3 with probabilities 1/4, 1/4 and 1/2, the value assigned to is 1/4 + 1/2 = 3/4, as in the diagram on the right. The conditional probability based on the intersection of events defined as: P = P P. satisfies the probability measure requirements so long as P is not zero. Probability measures are distinct from the more general notion of fuzzy measures in which there is no requirement that the fuzzy values sum up to 1, the additive property is replaced by an order relation based on set inclusion. Market measures which assign probabilities to financial market spaces based on actual market movements are examples of probability measures which are of interest in mathematical finance, e.g. in the pricing of financial derivatives. For instance, a risk-neutral measure is a probability measure which assumes that the current value of assets is the expected value of the future payoff taken with respect to that same risk neutral measure, discounted at the risk-free rate.
If there is a unique probability measure that must be used to price assets in a market the market is called a complete market. Not all measures that intuitively represent chance or likelihood are probability measures. For instance, although the fundamental concept of a system in statistical mechanics is a measure space, such measures are not always probability measures. In general, in statistical physics, if we consider sentences of the form "the probability of a system S assuming state A is p" the geometry of the system does not always lead to the definition of a probability measure under congruence, although it may do so in the case of systems with just one degree of freedom. Probability measures are used in mathematical biology. For instance, in comparative sequence analysis a probability measure may be defined for the likelihood that a variant may be permissible for an amino acid in a sequence. Borel measure Fuzzy measure Haar measure Martingale measure Lebesgue measure Probability and Measure by Patrick Billingsley, 1995 John Wiley ISBN 978-0-471-00710-4 Probability & Measure Theory by Robert B.
Ash, Catherine A. Doléans-Dade 1999 Academic Press ISBN 0-12-065202-1. Media related to Probability measure at Wikimedia Commons