---
tag: GEA8H3228S
kind: definition
name: Integral of a non-negative function
msc: [28A25]
defines: DG76G5Z6XW
uses: [P15VQS5DJ9, YYNCJSR0PJ, 7WMBX7M4P6, 0BTMQ04556, 0GSZ7B0CGR]
sources:
  - url: https://en.wikipedia.org/wiki/Lebesgue_integral
---

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]].
Let $s = \sum_{i=1}^{n} a_i \mathbf{1}_{A_i}$ be a non-negative [[simple function|7WMBX7M4P6]], with $a_i \in [0, \infty)$ and pairwise disjoint $A_i \in \Sigma$.
Its integral is
$$\int_X s \, d\mu = \sum_{i=1}^{n} a_i \, \mu(A_i),$$
with the convention $0 \cdot \infty = 0$.
Let $f \colon X \to [0, \infty]$ be a [[measurable function|YYNCJSR0PJ]].
Its *integral* is the [[supremum|0BTMQ04556]], in the [[extended real line|0GSZ7B0CGR]],
$$\int_X f \, d\mu = \sup \set{\int_X s \, d\mu : s \text{ simple}, \ 0 \le s \le f}.$$

## Remarks

Example: let $X = \N$ with the counting measure and $f(n) = 2^{-n}$ for $n \ge 1$, $f(0) = 1$.
The simple functions $s_N = f \cdot \mathbf{1}_{\set{0, \dots, N}}$ lie below $f$, and their integrals $\sum_{n=0}^{N} 2^{-n}$ increase to $2$.
Every simple $s \le f$ has integral at most $\sum_n f(n) = 2$, so $\int f \, d\mu = 2$.

Non-example: the formula does not apply to $g(x) = \sin x$ on $\R$, because $g$ takes negative values.
Such functions need the splitting $g = g^+ - g^-$ of the Lebesgue integral.

The value of $\int_X s \, d\mu$ for a simple function does not depend on the way $s$ is written.
For $A \in \Sigma$ one writes $\int_A f \, d\mu = \int_X f \mathbf{1}_A \, d\mu$.
If $f = g$ almost everywhere, then $\int_X f \, d\mu = \int_X g \, d\mu$.