Kolmogorov axioms of probability

Axioms

Suppose sample space $\Omega$, event space $\mathcal{E}$, probability measure $P$, $P(E)$ probability of event $E \in \mathcal{E}$ (i.e., probability space $(\Omega, \mathcal{E},P)$). Then, hold the following axioms

1. $P(E) \geq 0$ for any $E \in \mathcal{E}$, $P(E) \in \mathbb{R}$
2. $P(\Omega) = 1$ (probability of universal set i.e. set of all outcomes is $1$)
3. If $A \cap B = \emptyset$ (i.e. events $A$ and $B$ are mutually exclusive), then $P(A \cup B) = P(A) + P(B)$
   • more generally (σ-additivity): probability of countable sequence of disjoint (mutually exclusive) events $E_1, E_2, ...$ is equal to the sum of probability of the individual events, i.e. $P\left( \bigcup_{i=1}^\infty E_i \right) = \sum_{i=1}^\infty P(E_i)$

References

1. Soch, Joram, et al. (2024). StatProofBook/StatProofBook.github.io: The Book of Statistical Proofs (Version 2023). Zenodo. https://doi.org/10.5281/ZENODO.4305949
   • https://statproofbook.github.io/D/prob-ax.html
2. https://en.wikipedia.org/wiki/Probability_axioms