## You are here

HomeKolmogorov zero-one law

## Primary tabs

# Kolmogorov zero-one law

# Kolomogorov zero-one law

###### 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
$\sigma(X_{1},X_{2},...,X_{n})$ ^{1}^{1}this 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
$\sigma(\bigcup_{{n=1}}^{{\infty}}\mathcal{F}_{n})$ ^{2}^{2}again by application
of the Monotone Class Theorem. But $\mathcal{F}_{{\infty}}\subset\sigma(\bigcup_{{n=1}}^{{\infty}}\mathcal{F}_{n})$ ^{3}^{3}because $\mathcal{F}_{{\infty}}\subset\sigma(X_{1},X_{2},...)=\sigma(\bigcup_{{n=1}}^{{%
\infty}}\mathcal{F}_{n})$,
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 $P(A)=1$.
∎

## Mathematics Subject Classification

28A05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier