---
tag: 04WSW8T3SY
kind: definition
name: Sigma-algebra
msc: [28A05]
defines: 0SV8K0TDGY
uses: [0SJ3RX9TXA, 0BC7KX51Z8, 0B1FSS1VZW, 07KVB4JSC7]
sources:
  - url: https://en.wikipedia.org/wiki/Σ-algebra
    locator: Definition
---

## Definition

Let $\Omega$ be a [[set|0SJ3RX9TXA]].
A σ-algebra on $\Omega$ is a subset $\mathcal{F}$ of the [[power set|0BC7KX51Z8]] of $\Omega$ with these three properties.
The set $\Omega$ is in $\mathcal{F}$.
If $A \in \mathcal{F}$, then its [[complement|0B1FSS1VZW]] $\Omega \setminus A$ is in $\mathcal{F}$.
If $A_1, A_2, A_3, \dots$ is a sequence of elements of $\mathcal{F}$, then the [[union|07KVB4JSC7]] $\bigcup_{n=1}^{\infty} A_n$ is in $\mathcal{F}$.
The pair $(\Omega, \mathcal{F})$ is a measurable space, and the elements of $\mathcal{F}$ are the measurable sets.

## Remarks

A σ-algebra contains $\varnothing$ and is closed under countable intersections, by De Morgan's laws.
The power set of $\Omega$ and the family $\{\varnothing, \Omega\}$ are σ-algebras on $\Omega$. They are the largest and the smallest σ-algebras on $\Omega$.
For each family $\mathcal{C}$ of subsets of $\Omega$, the intersection of all σ-algebras that contain $\mathcal{C}$ is a σ-algebra. It is the σ-algebra $\sigma(\mathcal{C})$ generated by $\mathcal{C}$.
The [[Borel σ-algebra|0BVHZXWKCD]] on $\mathbb{R}$ is the σ-algebra generated by the open intervals.

## Explanations

### L1 human

A σ-algebra is the list of the sets that we are allowed to measure.
If we can measure a set, then we can measure its complement.
If we can measure each set of a sequence, then we can measure their union.
For a finite sample space we can usually measure every subset, so the σ-algebra is the power set.