Kolmogorov zero-one law

Kolomogorov zero-one lawFernando Sanz Gamiz

Theorem (Kolmogorov).

Let $\mathrm{\Omega }$ be a set, $\mathrm{F}$ a sigma-algebra of subsets of $\mathrm{\Omega }$and $P$ a probability measure. Given the independent randomvariables $\mathrm{\{}{X}_n\mathrm{,}n\mathrm{\in }\mathrm{N}\mathrm{\}}$, defined on $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathrm{F}\mathrm{,}P\mathrm{)}$, ithappens that

$$P(A)=0\mathit{\text{ or }}P(A)=1,A\in {ℱ}_{\mathrm{\infty }},$$

i.e.,the probability of any tail event is 0 or 1.

Proof.

Define ${ℱ}_n=\sigma (X_1,X_2,\mathrm{\dots },X_n)$. As any event in$\sigma (X_{n+1},X_{n+2},\mathrm{\dots })$ is independent of any event in$\sigma (X_1,X_2,\mathrm{\dots },X_n)$ , anyevent in the tail $\sigma$-algebra ${ℱ}_{\mathrm{\infty }}$ is independent ofany event in ${\bigcup }_{n=1}^{\mathrm{\infty }}{ℱ}_n$; hence, any event in${ℱ}_{\mathrm{\infty }}$ is independent of any event in$\sigma ({\bigcup }_{n=1}^{\mathrm{\infty }}{ℱ}_n)$ . But ${ℱ}_{\mathrm{\infty }}\subset \sigma ({\bigcup }_{n=1}^{\mathrm{\infty }}{ℱ}_n)$ , so any tail event isindependent of itself, i.e., $P(A)=P(A\cap A)=P(A)P(A)$ whichimplies $P(A)=0$ or $P(A)=1$.∎