Let's clean up the "controversy": There's some pure math due to Lebesgue, Kolmogorov, and others that defines, clearly, precisely, a 'probability space' along

Let's clean up the "controversy": There's some pure math due to Lebesgue, Kolmogorov, and others that defines, clearly, precisely, a 'probability space' along with trials, events, probabilities, and random variables. It's a very nice box of math with a lot of really nice theorems with rock solid proofs, e.g., the algebra of random variables, approximation of random variables by other random variables, convergence of random variables, independence, conditioning and conditional independence, the central limit theorem, the weak and strong laws of large numbers, ergodic theory, Brownian motion, martingales and the martingale convergence theorem, Markov processes, and much more. That math is clean and rock solid. Does it work in practice? Often, yes. Much of the connection with practice is easy to see. For much of the rest, keys are when we have independence or conditional independence, and there are some surprisingly good, intuitive approaches to these two. Next, it works. Now, call it done. 'Controversy' over with. For the math, see, e.g., J. Neveu, 'Mathematical Foundations of the Calculus of Probability'. 'Frequentist'? That's essentially answered by the weak/strong laws of large numbers. Bayesian? That's essentially conditional probability, with the advanced approach via the Radon-Nikodym theorem with a famous proof by von Neumann. Can also prove some theorems that say that a Bayesian start doesn't have to hurt after a while.