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I am interested in a reference (article, maybe a book chapter) on the development of mathematical probability theory - that is, mostly starting from the beginning of the 20th century. It is surprising and fascinating to me that e.g. Wiener and Levy were able to develop theory of stochastic processes without using axiomatic probability theory. Similarly, I do remember seeing here on MO an article emphasizing that Kolmogorov's role in axiomatization of probability is usually overstated, whereas his job was mostly synthesis and the core ideas were developed by others.

I have recently read the paper by Shafer and Vovk "The origins and legacy of Kolmogorov’s Grundbegriffe" which more or less covers the topic until late 1930s. That is quite close to the reference I am looking for, however it does not say much about what happened after the Kolmogorov's book was published.

I have also tried reading Meyer's "Stochastic Processes from 1950 to present", however it is rather different from the first reference in focusing more on short reviews of specialized topics in theory of stochastic processes developed in 1950-2000, rather than discussing what were the important problems, how different people tried approaching them, which conflict of ideas were there, and why and how the ideas that are now in textbooks won over others (which is the scope of the first reference). In particular, Meyer just shortly comments on the definition of continuous-time stochastic processes: Doob found the Kolmogorov's existence theorem insufficient for his purposes as product $\sigma$-algebra was too scarce, and came up with something different, without giving the details of how it happened and what was the line of reasoning of Doob and others that time.

On a related note, although the first reference covers contributions of Borel and Lebesgue to abstract measure theory, as much as that of others (Caratheodory, Radon etc.), it does it mostly from the point of view of importance for Kolmogorov's axiomatization of probability. I'm thus interested in finding a similar account for the history of measure theory until the moment it was used in probability, and how was it affected in a feedback from probability after Kolmogorov, Doob and others started using it to solve probabilistic problems.

I would also appreciated an advice whether to leave this question here, or move it to MO.

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