---
tag: 0BVHZXWKCD
kind: definition
name: Borel sigma-algebra
msc: [28A05]
defines: 0WQXY32Q8D
uses: [04WSW8T3SY, 08J2M1JYAG, 0TK8Y3J5RW]
sources:
  - url: https://en.wikipedia.org/wiki/Borel_set
    locator: Generating the Borel algebra
---

## Definition

Let $X$ be a topological space.
The Borel σ-algebra $\mathcal{B}(X)$ is the smallest [[σ-algebra|04WSW8T3SY]] on $X$ that contains each open subset of $X$. It is the intersection of all σ-algebras on $X$ that contain the open sets.
A Borel set of $X$ is an element of $\mathcal{B}(X)$.

## Remarks

On the [[real line|08J2M1JYAG]], $\mathcal{B}(\mathbb{R})$ is also the σ-algebra generated by the open [[intervals|0TK8Y3J5RW]] $(a, b)$. The half-lines $(-\infty, b]$ with $b \in \mathbb{R}$ also generate it.
Each open set, each closed set, each countable set and each interval of $\mathbb{R}$ is a Borel set.
Some subsets of $\mathbb{R}$ are not Borel sets. For example, the Vitali sets are not Lebesgue measurable, so they are not Borel sets.

## Explanations

### L1 human

We want to give a probability to intervals such as "the waiting time is between $2$ and $5$ minutes".
The Borel sets are all the sets that we can build from intervals with complements and countable unions, again and again. So they are the events that come from intervals.