---
tag: 073Q8Z94KG
kind: statement
role: theorem
name: Kolmogorov's zero–one law
aliases: [kolmogorov-0-1-law, kolmogorov-zero-one-law]
msc: [60F20]
uses: [0XCE070V8B, 09T8ZHST1V, 04WSW8T3SY]
sources:
  - url: https://en.wikipedia.org/wiki/Kolmogorov's_zero–one_law
    locator: Formulation
same_as:
  wikidata: Q2004664
  wikipedia: Kolmogorov's zero–one law
---

## Statement

Let $X_1, X_2, \dots$ be [[independent|0XCE070V8B]] random variables on a probability space $(\Omega, \mathcal{F}, P)$.
Let $\mathcal{T} = \bigcap_{n=1}^{\infty} \sigma(X_n, X_{n+1}, \dots)$ be the tail σ-algebra, where $\sigma(X_n, X_{n+1}, \dots)$ is the smallest [[σ-algebra|04WSW8T3SY]] that makes $X_m$ measurable for all $m \ge n$.
Then $P(A) \in \{0, 1\}$ for each $A \in \mathcal{T}$.

## Remarks

An event is in $\mathcal{T}$ if changing finitely many of the $X_n$ cannot change whether it occurs.
Examples of tail events: "the series $\sum_n X_n$ converges" and "$\limsup_n (X_1 + \dots + X_n)/n \le c$" for a real $c$.
Proof idea: $\mathcal{T}$ is [[independent|09T8ZHST1V]] of $\sigma(X_1, \dots, X_n)$ for each $n$, hence of $\sigma(X_1, X_2, \dots)$, which contains $\mathcal{T}$. So each $A \in \mathcal{T}$ is independent of itself, and $P(A) = P(A)^2$.
The theorem says that a tail event is almost sure or almost impossible. It does not say which of the two cases holds.

## History

The theorem is named after Andrey Kolmogorov.

## Explanations

### L1 human

Some events depend only on the "far end" of an infinite sequence of independent trials, such as "the averages converge".
For such an event, the probability cannot be $1/2$ or any value strictly between $0$ and $1$: it is always $0$ or $1$.