The provided sources detail the measure-theoretic foundations of modern probability theory, a rigorous mathematical framework established primarily by Andrey Kolmogorov in 1933. This framework replaced heuristic approaches to probability to resolve paradoxes associated with continuous and infinite sample spaces.

Here is a brief explanation of the core concepts covered in the texts:

• Kolmogorov's Axioms: Probability is formalized using a probability space $(\Omega, \mathcal{F}, P)$. Here, $\Omega$ is the sample space (all possible outcomes), $\mathcal{F}$ is a $\sigma$-algebra (a collection of events closed under countable unions and complements), and $P$ is a probability measure. $P$ must satisfy three axioms: non-negativity, normalization ($P(\Omega) = 1$), and countable additivity (the probability of a union of mutually exclusiv ...