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I am working on a term paper for an analysis course and I thought it would be interesting to talk about the connection between analysis and probability theory. Honestly, it would also benefit me a lot as a student in statistics. Knowing the historical context of the things I'm learning about would help me better understand their significance.

I vaguely remember from some of my probability courses that people have discovered the intuition of law of large numbers and central limit theorem very early, yet early attempts to put it on rigorous mathematical foundation failed, until the development of measure theory, Lebesgue integration, Fourier transformation, and so on.

I am wondering whether there are some nice books/review papers that summarize this process. An ideal reference would include something like, someone tried to prove the law of large numbers, yet failed, since the integration theory back then wasn't adequate, someone tried doing the central limit theory and came up with the characteristic function idea, yet the one-to-one mapping between random variables and characteristic functions couldn't be established, due to some other lack of math tools back then. Finally Kolmogorov came into the picture, put together a complete set of tools and finished the job, by filling in something and something some earlier mathematicians missed.

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I don't know if this counts as an answer, or just a comment (moderators please!), but I believe one should look to the works of Anders Hald.

A History of Parametric Statistical Inference from Bernoulli to Fisher, 1713–1935 2007 Springer

A History of Probability and Statistics and Their Applications before 1750 (2003) Wiley

A History of Mathematical Statistics from 1750 to 1930 (1998).Wiley

ADDENDUM
The question was cross-posted to math.SE, but given its nature, I believe this time cross-posting had a beneficial effect - the books suggested by the other answer are totally different, see https://math.stackexchange.com/questions/1030229/what-are-some-good-references-on-how-probability-theory-got-mathematically-rigor

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