When were infinite probability spaces first rigorously dealt with?

Question

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.

Answer 1

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.