Independent Random Variables

Statements

Definition. Two random variables $X_1:\Omega \to {\mathcal{X}}_1$ and $X_2:\Omega \to {\mathcal{X}}_2$ are said to be independent, if for every $S_1\subseteq {\mathcal{X}}_1$ and $S_2\subseteq {\mathcal{X}}_2$ the following equality

$P(X_1\in S_1,X_2\in S_2)=P(X_1\in S_1)P(X_2\in S_2)$

holds.

References

Ost20, Definition 5.4.10