TITLE: What distinguishes the Measure Theory and Probability Theory?
QUESTION [19 upvotes]: It is clear that the Theory of Probability works primarily with limited measures on measurable spaces.
On the other hand there is a folklore that says that what distinguishes Measure Theory and Probability Theory is the conditional probability and conditional expectation.
But conditional probabilities and conditional expectations are derived from $\,$ Radon–Nikodym theorem and a measure with respect to another measure. And The Radon–Nikodym theorem is a typical result of Measure Theory.

Question 1: So we could see the Theory of Probability as a subdiscipline of Measure Theory?

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Question 2: It would be possible to give another basis for the Theory of Probability than not the Theory of Measure?

REPLY [3 votes]: Salomon Bochner's words in the introduction to his article entitled 'Stochastic Processes' published in 1947 in volume 48 of the Annals of Matematics: