# Kolmogorov zero-one law

Kolomogorov zero-one lawFernando Sanz Gamiz

###### Theorem (Kolmogorov).

Let $\Omega$ be a set, $\mathcal{F}$ a sigma-algebra of subsets of $\Omega$ and $P$ a probability measure. Given the independent random variables $\{X_{n},n\in\mathbb{N}\}$, defined on $(\Omega,\mathcal{F},P)$, it happens that

 $P(A)=0\mbox{ or }P(A)=1,A\in\mathcal{F}_{\infty},$

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

###### Proof.

Define $\mathcal{F}_{n}=\sigma(X_{1},X_{2},...,X_{n})$. As any event in $\sigma(X_{n+1},X_{n+2},...)$ is independent of any event in $\sigma(X_{1},X_{2},...,X_{n})$ 11this assertion should be proved actually, because independence of random variables is defined for every finite number of them and we are dealing with events involving an infinite number. By two successive applications of the Monotone Class Theorem, one can readily prove this is in fact correct, any event in the tail $\sigma$-algebra $\mathcal{F}_{\infty}$ is independent of any event in $\bigcup_{n=1}^{\infty}\mathcal{F}_{n}$; hence, any event in $\mathcal{F}_{\infty}$ is independent of any event in $\sigma(\bigcup_{n=1}^{\infty}\mathcal{F}_{n})$ 22again by application of the Monotone Class Theorem. But $\mathcal{F}_{\infty}\subset\sigma(\bigcup_{n=1}^{\infty}\mathcal{F}_{n})$ 33because $\mathcal{F}_{\infty}\subset\sigma(X_{1},X_{2},...)=\sigma(\bigcup_{n=1}^{% \infty}\mathcal{F}_{n})$, this last equality being easily proved, so any tail event is independent of itself, i.e., $P(A)=P(A\cap A)=P(A)P(A)$ which implies $P(A)=0$ or $P(A)=1$. ∎

Title Kolmogorov zero-one law KolmogorovZerooneLaw 2013-03-22 17:07:21 2013-03-22 17:07:21 fernsanz (8869) fernsanz (8869) 10 fernsanz (8869) Definition msc 28A05 TailEvent