TITLE: When were infinite probability spaces first rigorously dealt with?
QUESTION [5 upvotes]: In other words, when was the transition from exclusive focus on Classical Probability Theory effected? I’m interested in correlating this (approximate) date with something else I am working on, and it would be interesting to know in any case.

REPLY [1 votes]: Borel, É. (1909). "Les probabilités dénombrables et leurs applications arithmétiques", Rendiconti del Circolo Matematico di Palermo 27, 1055-1079
is usually credited for the introduction of $\sigma$-additivity and for the construction of random variables as measurable functions.
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