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Foundations of the Theory of Probability
This famous little book was first published in German in 1933 and in Russian a few years later, setting forth the axiomatic foundations of modern probability theory and cementing the author's reputation as a leading authority in the field. The distinguished Russian mathematician A. N. Kolmogorov wrote this foundational text, and it remains important both to students beginning a serious study of the topic and to historians of modern mathematics.
Suitable as a text for advanced undergraduates and graduate students in mathematics, the treatment begins with an introduction to the elementary theory of probability and infinite probability fields. Subsequent chapters explore random variables, mathematical expectations, and conditional probabilities and mathematical expectations. The book concludes with a chapter on the law of large numbers, an Appendix on zero-or-one in the theory of probability, and detailed bibliographies.
Suitable as a text for advanced undergraduates and graduate students in mathematics, the treatment begins with an introduction to the elementary theory of probability and infinite probability fields. Subsequent chapters explore random variables, mathematical expectations, and conditional probabilities and mathematical expectations. The book concludes with a chapter on the law of large numbers, an Appendix on zero-or-one in the theory of probability, and detailed bibliographies.
96 pages, Paperback
First published January 1, 1933
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About the author
A.N. Kolmogorov
41 books21 followersDr. Andrey Nikolaevich Kolmogorov, Ph.D. (Moscow State University, 1929; Russian: Андре́й Никола́евич Колмого́ров) was a Soviet mathematician and professor at the Moscow State University where he became the first chairman of the department of probability theory two years after the 1933 publication of his book which laid the modern axiomatic foundations of the field. He was a Member of the Russian Academy of Sciences and winner of many awards, including the Stalin Prize (1941), the Lenin Prize (1965), the Wolf Prize (1980), and the Lobachevsky Prize (1986).
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Displaying 1 - 9 of 9 reviews
June 28, 2022
The theory of probability is all but trivial in finite state spaces, such as are typical of cards and games of chance. One must suppose the field had reached a state of maturity by the time of Laplace’s essay of 1819, at the latest. When indefinite sequences of coin tosses, say, are brought into consideration, the theoretical and speculative issues become rather subtle, as in the investigations of von Mises into the nature of true randomness. The same holds for the continuous state spaces typical of statistical physics, where one must decide what classes of sets to entertain as the elementary events.
Kolmogorov’s work, dating originally to 1933, revolutionized the field. By the 1950’s, his approach had become conventional among probabilists, and Doob, for instance, treats Kolmogorov’s axioms as unproblematical, but to appreciate the significance of Kolmogorov’s contribution one must go back to the earlier period, when he was controversial. From a formal point of view, the sole axiom of significance is the last, postulating countable subadditivity of the probability measure (which matters, of course, only in the case of infinite state spaces). Here, Kolmogorov’s intuition seems to have seized upon the key property that renders measure theory, as we know it since the early twentieth century, really effective for mathematicians who must reason discursively. For, even supposing—contra Kant—the human mind capable of an intellectual intuition, it certainly lacks the facile access to it characteristic of the angelic intelligences. In this recensionist’s opinion, all of the power of measure theory in real analysis derives from the condition of countable subadditivity, which makes relevant the concept of sigma-algebras. The flourishing of probability theory in the post-Second-World-War period—with the development of the Wiener process for Brownian motion, characterization of Markovian processes and the theory of martingales and semi-martingales etc., all with non-trivial applications to partial differential equations such as the diffusion equation—must be attributed to the theorem-proving potential unleashed by Kolmogorov’s clever surmise about the appropriate axiomatic foundations.
The slenderness of Kolmogorov’s little treatise—at just 70 pages—belies the wealth of applications it was to make possible. Kolmogorov’s text itself treats little beyond the elementary definitions than formal statements, such as necessary and sufficient criteria for convergence of sequences of random variables in various spaces or the introduction of conditional expectations, without any deep results or substantial applications of the ideas at all, although the zero-or-one law (which states that under certain conditions, pairwise independence among them, sequences of random variables must either converge or diverge almost surely) is of some theoretical interest. For instance, Kolmogorov shows how to construct a sensible probability measure in infinite-dimensional space but fails to give any examples at all. One had to wait a decade until Wiener produced his process for Brownian motion, inessential variants of which all but exhaust the range of known non-trivial cases of measures in infinitely many dimensions (such as the Ornstein-Uhlenbeck processes in non-equilibrium statistical mechanics or the Feynman-Kac integral for interacting fields in constructive quantum field theory).
Thus, one could say that for most mathematicians and physicists, Kolmogorov reduced probability theory to real analysis on measure spaces of unit mass. The charm of this little volume, though, is that that was not a foregone conclusion at the time of its appearance. In a section barely more than a page in length, Kolmogorov, in an elliptical reference to, inter alia, Keynes and von Mises, points out that his formalization is indeed debatable from a philosophical point of view. Apart from this, his presentation is as dry as can be. One wonders at the price that has been paid. Just as Descartes’ methodological elimination of teleological causes paved the way to the astonishing successes of the mechanistic world-view, Kolmogorov’s axiomatization made possible the remarkable post-war expansion of probability theory at the cost of suppressing speculation on the real meaning of probability, which is—for this recensionist—a vital topic in today’s world. For probability functions in a very different manner in the real world, containing as it does human agents, than it does in the world of physics and natural science, where the initial conditions are indifferent. Nobody is willing to be indifferent when it comes to human affairs, though, when his fate is at stake.
Kolmogorov’s work, dating originally to 1933, revolutionized the field. By the 1950’s, his approach had become conventional among probabilists, and Doob, for instance, treats Kolmogorov’s axioms as unproblematical, but to appreciate the significance of Kolmogorov’s contribution one must go back to the earlier period, when he was controversial. From a formal point of view, the sole axiom of significance is the last, postulating countable subadditivity of the probability measure (which matters, of course, only in the case of infinite state spaces). Here, Kolmogorov’s intuition seems to have seized upon the key property that renders measure theory, as we know it since the early twentieth century, really effective for mathematicians who must reason discursively. For, even supposing—contra Kant—the human mind capable of an intellectual intuition, it certainly lacks the facile access to it characteristic of the angelic intelligences. In this recensionist’s opinion, all of the power of measure theory in real analysis derives from the condition of countable subadditivity, which makes relevant the concept of sigma-algebras. The flourishing of probability theory in the post-Second-World-War period—with the development of the Wiener process for Brownian motion, characterization of Markovian processes and the theory of martingales and semi-martingales etc., all with non-trivial applications to partial differential equations such as the diffusion equation—must be attributed to the theorem-proving potential unleashed by Kolmogorov’s clever surmise about the appropriate axiomatic foundations.
The slenderness of Kolmogorov’s little treatise—at just 70 pages—belies the wealth of applications it was to make possible. Kolmogorov’s text itself treats little beyond the elementary definitions than formal statements, such as necessary and sufficient criteria for convergence of sequences of random variables in various spaces or the introduction of conditional expectations, without any deep results or substantial applications of the ideas at all, although the zero-or-one law (which states that under certain conditions, pairwise independence among them, sequences of random variables must either converge or diverge almost surely) is of some theoretical interest. For instance, Kolmogorov shows how to construct a sensible probability measure in infinite-dimensional space but fails to give any examples at all. One had to wait a decade until Wiener produced his process for Brownian motion, inessential variants of which all but exhaust the range of known non-trivial cases of measures in infinitely many dimensions (such as the Ornstein-Uhlenbeck processes in non-equilibrium statistical mechanics or the Feynman-Kac integral for interacting fields in constructive quantum field theory).
Thus, one could say that for most mathematicians and physicists, Kolmogorov reduced probability theory to real analysis on measure spaces of unit mass. The charm of this little volume, though, is that that was not a foregone conclusion at the time of its appearance. In a section barely more than a page in length, Kolmogorov, in an elliptical reference to, inter alia, Keynes and von Mises, points out that his formalization is indeed debatable from a philosophical point of view. Apart from this, his presentation is as dry as can be. One wonders at the price that has been paid. Just as Descartes’ methodological elimination of teleological causes paved the way to the astonishing successes of the mechanistic world-view, Kolmogorov’s axiomatization made possible the remarkable post-war expansion of probability theory at the cost of suppressing speculation on the real meaning of probability, which is—for this recensionist—a vital topic in today’s world. For probability functions in a very different manner in the real world, containing as it does human agents, than it does in the world of physics and natural science, where the initial conditions are indifferent. Nobody is willing to be indifferent when it comes to human affairs, though, when his fate is at stake.
September 8, 2024
Probability is a field that is sometimes counterintuitive. This monograph focuses on the various aspects of Probability and how it works out. The book itself is really short, comprising only six chapters. I can even list them; Elementary Theory of Probability, Infinite Probability Fields, Random Variables, Mathematical Expectations, Conditional Probabilities and Mathematical Expectations, and Independence: The Law of Large Numbers.
The book utilizes Lebesgue’s theories of measure and integration. If you know what that means, you have a leg up on me. I still haven’t advanced beyond Calculus II.
The author is Professor A N Kolmogorov. I had heard of him before, but only to the extent of reading other books in this field. I don’t know how well perceived he is in the field of Probability.
There isn’t much else to say about this book, but it does have a bibliography and a preface.
The book utilizes Lebesgue’s theories of measure and integration. If you know what that means, you have a leg up on me. I still haven’t advanced beyond Calculus II.
The author is Professor A N Kolmogorov. I had heard of him before, but only to the extent of reading other books in this field. I don’t know how well perceived he is in the field of Probability.
There isn’t much else to say about this book, but it does have a bibliography and a preface.
August 5, 2018
We are in 1933, Kolmogorov is writing down the axioms of probability theory. Just five neat axioms. From these five, he goes about listing the corollaries and theorems (e.g. Bayes) defining probability theory. He is sharp going straight to the point without examples and exercises. As he moves forward, things get more and more abstract; e.g. adding an axiom for continuity so he can define probability on infinite fields. This is a short and elegant example of consistent reasoning.
December 29, 2019
I really liked the way the basic concepts of probability was introduced early on. However, I found the exposition about more involved topics harder due to either translation issues (e.g., what is "complex C of conditions"?) or terseness of presentation. That said, not a light read.
Want to read
December 8, 2022AMP and Steenbock libraries, Library-of-Congress QA273 K614 1956
June 19, 2021
We have already reviewed Kolmogoroff’s foundational 1932 contribution to twentieth-century probability theory in English translation, here. The present review is premised on a thought-experiment: what more can one get out of Kolmogoroff by going back to the German original? A common view that has much to speak for it would be this, that the more poetic in nature a work is, the more that is lost in translation, whereas the more scientific, the less. Will it hold up when put to the test?
One notices at once upon venturing into the text, that the concept of measure depends on an arbitrary constant:
Zweck des vorliegenden Heftes ist eine axiomatische Begründung der Wahrscheinlichkeitsrechnung. Der leitende Gedanke des Verfassers war dabei, die Grundbegriffe der Wahrscheinlichkeitsrechnung, welche noch unlängst für ganz eigenartig galten, natürlicherweise in die Reihe der allgemeinen Begriffsbildungen der modernen Mathematik einzuordnen. Vor der Entstehung der Lebesgueschen Maß- und Integrationstheorie war diese Aufgabe ziemlich hoffnungslos. Nach den Lebesgueschen Untersuchungen lag die Analogie zwichen dem Maße einer Menge und der Wahrscheinlichkeit eines Ereignisses sowie zwischen dem Integral einer Funktion und der mathematischen Erwartung einer zufälligen Größe auf der Hand. Diese Analogie ließ sich auch weiter fortführen….Um, ausgehend von dieser Analogie, die Wahrscheinlichkeitsrechnung zu begründen, hätte man noch die Maß und Integrationstheorie von den geometrischen Elementen, welche bei Lebesgue noch hervortreten, zu liefern. Diese Befreiung wurde von Fréchet vollzogen. (p. 1)
The first philosophical question to emerge, then, is that as to what may be the relation between measure and probability, as ordinarily conceived? We can catch a glimpse of what is at stake in the continuum case; roughly: what if something could happen in the infinitesimal spaces between the points of a continuous manifold? Kolmogorov remarks, in this connection, on the importance of freedom of choice behind the axioms of measure theory:
Die Axiomatisierung der Wahrscheinlichkeitsrechnung kann auf verschiedene Weisen geschehen, und zwar beziehen sich diese verschiedenen Möglichkeiten sowohl auf die Wahl der Axiome als auch auf die der Grundbegriffe und Grundrelationen. Wenn man allerdings das Ziel der möglichsten Einfachheit des Axiomssystems und des weiteren Aufbaus der darauf folgenden Theorie im Auge hat, so scheint es am zweckmäßigsten, die Begriffe eines zufälligen Ereignisses und seiner Wahrscheinlichkeit zu axiomatisieren. (p. 2)
The point is that, for instance, the construction of level sets of the expectation would be different in infinitely many dimensions or higher order. A complex of conditions partitions a box into non-overlapping cells. Conditional probabilities would be affected in an entirely different way if the measure function were to fail to be additive. Nobody considers any such, but it is clear that non-additive characteristics could be key to integration in controverted cases.
Conclusion: it can prove most stimulating to consult a mathematical text in the original! Novalis calls the phenomenon Blütenstäube:
Geistvoll ist das, worin sich der Geist unaufhörlich offenbart – wenigstens oft von neuem, in veränderter Gestalt wieder erscheint – nicht bloß etwa nur einmal – so im Anfang – wie bei vielen philosophischen Systemen. (Nr. 31)
Wir sind auf einer Mission. Zur Bildung der Erde sind wir berufen. Wenn uns ein Geist erschiene, so würden wir uns sogleich unsrer eignen Geistigkeit bemächtigen – wir würden inspiriert sein, durch uns und den Geist zugleich – ohne Inspiration keine Geistererscheinung. Inspiration ist Erscheinung und Gegenerscheinung, Zueignung und Mitteilung zugleich. (Nr. 32)
Jetzt regt sich nur hie und da Geist – wenn wird der Geist sich im Ganzen regen? – wenn wird die Menschheit in Masse sich selbst zu besinnen anfangen? (Nr. 37)
One notices at once upon venturing into the text, that the concept of measure depends on an arbitrary constant:
Zweck des vorliegenden Heftes ist eine axiomatische Begründung der Wahrscheinlichkeitsrechnung. Der leitende Gedanke des Verfassers war dabei, die Grundbegriffe der Wahrscheinlichkeitsrechnung, welche noch unlängst für ganz eigenartig galten, natürlicherweise in die Reihe der allgemeinen Begriffsbildungen der modernen Mathematik einzuordnen. Vor der Entstehung der Lebesgueschen Maß- und Integrationstheorie war diese Aufgabe ziemlich hoffnungslos. Nach den Lebesgueschen Untersuchungen lag die Analogie zwichen dem Maße einer Menge und der Wahrscheinlichkeit eines Ereignisses sowie zwischen dem Integral einer Funktion und der mathematischen Erwartung einer zufälligen Größe auf der Hand. Diese Analogie ließ sich auch weiter fortführen….Um, ausgehend von dieser Analogie, die Wahrscheinlichkeitsrechnung zu begründen, hätte man noch die Maß und Integrationstheorie von den geometrischen Elementen, welche bei Lebesgue noch hervortreten, zu liefern. Diese Befreiung wurde von Fréchet vollzogen. (p. 1)
The first philosophical question to emerge, then, is that as to what may be the relation between measure and probability, as ordinarily conceived? We can catch a glimpse of what is at stake in the continuum case; roughly: what if something could happen in the infinitesimal spaces between the points of a continuous manifold? Kolmogorov remarks, in this connection, on the importance of freedom of choice behind the axioms of measure theory:
Die Axiomatisierung der Wahrscheinlichkeitsrechnung kann auf verschiedene Weisen geschehen, und zwar beziehen sich diese verschiedenen Möglichkeiten sowohl auf die Wahl der Axiome als auch auf die der Grundbegriffe und Grundrelationen. Wenn man allerdings das Ziel der möglichsten Einfachheit des Axiomssystems und des weiteren Aufbaus der darauf folgenden Theorie im Auge hat, so scheint es am zweckmäßigsten, die Begriffe eines zufälligen Ereignisses und seiner Wahrscheinlichkeit zu axiomatisieren. (p. 2)
The point is that, for instance, the construction of level sets of the expectation would be different in infinitely many dimensions or higher order. A complex of conditions partitions a box into non-overlapping cells. Conditional probabilities would be affected in an entirely different way if the measure function were to fail to be additive. Nobody considers any such, but it is clear that non-additive characteristics could be key to integration in controverted cases.
Conclusion: it can prove most stimulating to consult a mathematical text in the original! Novalis calls the phenomenon Blütenstäube:
Geistvoll ist das, worin sich der Geist unaufhörlich offenbart – wenigstens oft von neuem, in veränderter Gestalt wieder erscheint – nicht bloß etwa nur einmal – so im Anfang – wie bei vielen philosophischen Systemen. (Nr. 31)
Wir sind auf einer Mission. Zur Bildung der Erde sind wir berufen. Wenn uns ein Geist erschiene, so würden wir uns sogleich unsrer eignen Geistigkeit bemächtigen – wir würden inspiriert sein, durch uns und den Geist zugleich – ohne Inspiration keine Geistererscheinung. Inspiration ist Erscheinung und Gegenerscheinung, Zueignung und Mitteilung zugleich. (Nr. 32)
Jetzt regt sich nur hie und da Geist – wenn wird der Geist sich im Ganzen regen? – wenn wird die Menschheit in Masse sich selbst zu besinnen anfangen? (Nr. 37)
March 4, 2023
Short little treatise on probability, 70 pages, that built probability form set and measure theory. A classic in the field, and fairly readable for a mathematics text.
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December 21, 2016fumate un porro antes
February 10, 2023
An historical document elaborating over the evolving comprehension of probability theory in the early 2oth century.
Displaying 1 - 9 of 9 reviews