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The development of rigour in mathematical probability, (1900–1950). (English) Zbl 0806.01014
Pier, Jean-Paul (ed.), Development of mathematics 1900-1950. Based on a symposium organized by the Luxembourg Mathematical Society in June 1992, at Château Bourglinster, Luxembourg. Basel: Birkhäuser. 157-170 (1994).
The history of logical development of mathematical probability took a slow and zigzag-form course. Full acceptance of mathematical probability was not realized until the second half of the century. There exist close connections to potential theory (strong subadditivity inequality, G. A. A. Choquet’s theory of mathematical capacity), to the theory of Brownian motion (K. Ito 1944), and to martingales (Ville 1939), already in 1900 Louis J. B. A. Bachelier derived various important distributions related to the Brownian motion process. Measure theory started with Henri Lebesgue’s thesis (1902) which extended the definition of volume \(R^ N\) to the Borel sets. Johann K. A. Radon (1913) generalized measures of Borel sets of \(R^ N\) by completion. Maurice René Fréchet (1915) defined a positive countable additive set function, Percy J. Daniell (1918) measures in infinite dimensional Euclidean space. The Radon- Nikodym theorem (1930) shows that this can be expressed as an integral over the sets. Thus it lasted 28 years before Lebesgue’s theory was extended far enough to be adequate for the mathematical basis of mathematical probability. New relations between functions were made possible by the mathematization of probability (S. N. Bernstein 1927). A. N. Kolmogorov (1933) constructed the strong basis for mathematical probability. The influence of the present, given the past, depends only on the immediate past. Markov property, introduced by a very special case by A. A. Markov in 1906 has proved very fruitful. Many of the most essential results of mathematical probability have been suggested by nonmathematical context of real world probability, which has been had never even a universally accepted definition. In fact the relation between real world probability and mathematical probability has been simultaneously the bane of and inspiration for the development of mathematical probability. The difficulty in separating mathematical probability from the rest of analysis, the measure space is a probability space but with trivial changes the discussion is valid for any finite measure space.
For the entire collection see [Zbl 0796.00016].
Reviewer: H.Grimm (Jena)

01A60 History of mathematics in the 20th century
60-03 History of probability theory
60A10 Probabilistic measure theory
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures