---
tag: AEX1QP3178
kind: definition
name: Baire function
msc: [26A21]
defines: K26WG8C6PV
uses: [063W5JMY85, 0P27W9RW0G, 056FSKG32J, 0RDJK7EQYY, 0T4EJ8ZJM9, 0F1WR9C83K, 0BVHZXWKCD]
sources:
  - url: https://en.wikipedia.org/wiki/Baire_function
    locator: Classification of Baire functions
---

## Definition

Let $X$ be a [[metric space|063W5JMY85]], for example a subset of $\R$.
Let $\omega_1$ be the first uncountable [[ordinal|0RDJK7EQYY]].
For each ordinal $\alpha < \omega_1$, define a set $B_\alpha$ of functions $X \to \R$ as follows.

1. $B_0$ is the set of [[continuous functions|0P27W9RW0G]] $X \to \R$.
2. For $0 < \alpha < \omega_1$, $B_\alpha$ is the set of functions $f \colon X \to \R$ that are [[pointwise limits|056FSKG32J]] $f(x) = \lim_{n \to \infty} f_n(x)$ of sequences $(f_n)$ in $\bigcup_{\beta < \alpha} B_\beta$.

A function of *Baire class* $\alpha$ is an element of $B_\alpha$.
A *Baire function* is a function that lies in $B_\alpha$ for some $\alpha < \omega_1$.

## Remarks

The classes increase: $B_\beta \subseteq B_\alpha$ for $\beta \le \alpha$, since a function is the limit of a constant sequence.
The set of Baire functions is the smallest set of functions $X \to \R$ that contains the continuous functions and is closed under pointwise limits of sequences.
For $X = \R$, the Baire functions are exactly the functions that are measurable for the [[Borel sigma-algebra|0BVHZXWKCD]].

Example: if $F \colon \R \to \R$ is differentiable, then $F'$ is of Baire class $1$.
Indeed $F'(x) = \lim_{n \to \infty} n \big(F(x + 1/n) - F(x)\big)$, and each function on the right is continuous.

Non-example: the [[Dirichlet function|0T4EJ8ZJM9]] $\mathbf 1_\Q$ is not of Baire class $1$.
A function of Baire class $1$ on $\R$ is continuous at the points of a dense set, while $\mathbf 1_\Q$ is continuous nowhere.
It is of Baire class $2$, since $\mathbf 1_\Q(x) = \lim_{m \to \infty} \lim_{n \to \infty} \cos(m! \pi x)^{2n}$.