History: Probability Theory
Of course they're both major oversimplifications, but which of (1) and (2) is closer to the truth?
Lebesgue invents measure theory and then Kolmogorov notices that measure theory can be used to axiomatize probability theory.
Lebesgue invents measure theory, Kolmogorov gives an axiomatization of probability theory, then someone notices the connection.
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."
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.