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Of course they're both major oversimplifications, but which of (1) and (2) is closer to the truth?

  1. Lebesgue invents measure theory and then Kolmogorov notices that measure theory can be used to axiomatize probability theory.

  2. Lebesgue invents measure theory, Kolmogorov gives an axiomatization of probability theory, then someone notices the connection.

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    $\begingroup$ What research have you done? In particular, have you read any of the biographies of Kolmogorov available online? $\endgroup$ – marty cohen Nov 4 '15 at 22:53
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    $\begingroup$ Try History of Science and Mathematics S.E. :) $\endgroup$ – Megadeth Nov 4 '15 at 22:57
  • $\begingroup$ Two excellent suggestions. Didn't know there was a HSMSE. I've seen plenty of statements about the origin of probability theory which I just realized didn't actually clarify this question - never thought of looking at a biography, duh. $\endgroup$ – David C. Ullrich Nov 4 '15 at 23:09
  • $\begingroup$ @martycohen I've looked at four such biographies. Nothing so far really answers the question. I find statements like "In 1933, Kolmogorov published his book, Foundations of the Theory of Probability, laying the modern axiomatic foundations of probability theory and establishing his reputation as the world's leading expert in this field", which is consistent with either (1) or (2). $\endgroup$ – David C. Ullrich Nov 4 '15 at 23:15
  • $\begingroup$ @DavidC.Ullrich Okay, great. As you wish. Now deleted on HSM. $\endgroup$ – HDE 226868 Nov 5 '15 at 1:09
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From the Preface to Kolmogorov's 1933 book:

"The purpose of this monograph is to give an axiomatic foundation for the theory of probability. The author set himself the task of putting in their natural place, among the general notions of modern mathematics, the basic concepts of probability theory -- concepts which until recently were considered to be quite peculiar.

This task would have been a rather hopeless one before the introduction of Lebesgue's theories of measure and integration. However, after Lebesgue's publication of his investigations, the analogies between measure of a set and probability of an event, and between integral of a function and mathematical expectation of a random variable, became apparent."

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  • $\begingroup$ Very good! What's the name of the book? $\endgroup$ – Integral Nov 4 '15 at 23:57
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    $\begingroup$ Nevermind, I just found. It's Foundations of the Theory of Probability. $\endgroup$ – Integral Nov 5 '15 at 0:01
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    $\begingroup$ "Foundations of the Theory of Probability" (in the original German "Grundbegriffe der Warscheinlichkeitrechnung"). $\endgroup$ – John Dawkins Nov 5 '15 at 0:02
  • $\begingroup$ I hate that rule saying we're not supposed to post comments saying nothing but "thanks". But rules are rules - I'm not even going to thank you for this. heh... $\endgroup$ – David C. Ullrich Nov 5 '15 at 0:03
  • $\begingroup$ All credits to Kolmogorov after all! This settles the question. $\endgroup$ – Integral Nov 5 '15 at 0:05
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Kolmogorov was neither the first to see a connection between measure theory and probability, nor did he claim to do so. Briefly after the part of the preface quoted in the answer by John Dawkins, Kolmogorov writes

While a conception of probability theory based on the above general viewpoints has been current for some time among certain mathematicians, there was lacking a complete exposition of the whole system, free of extraneous complication.

The most important mathematical contribution (besides the existence result for stochastic processes, essentially preceded by Daniell) of the thin book was giving a rigorous notion of conditional expectation based on the then very recent Radon-Nikodym theorem. But most of the book was a synthesis of previous work.

For the earlier work on measure theoretic probability, you should read The Sources of Kolmogorov’s Grundbegriffe by Glenn Shafer and Vladimir Vovk, an article that cleared up a lot of confusions I had.

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  • $\begingroup$ Thanks for clarifying that. So (1) is indeed an oversimplification, as it seemed it must be.... $\endgroup$ – David C. Ullrich Jul 15 '16 at 13:32

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