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.