Andrey Nikolaevich Kolmogorov

1. Kolmogorov Axioms

The foundation of modern probability theory.

$$ P(E) \ge 0, \quad P(\Omega) = 1, \quad P\left(\bigcup E_i\right) = \sum P(E_i) $$

2. Chapman-Kolmogorov Equation

Relates joint probability distributions of stochastic processes.

$$ p_{ij}(t+s) = \sum_k p_{ik}(t) p_{kj}(s) $$