pages/measure-and-integration/lebesgue-integral/almost-everywhere.md


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---
title: Almost Everywhere
parent: Lebesgue Integral
grand_parent: Measure and Integration
nav_order: 1
---

# {{ page.title }}

{% definition Almost Everywhere %}
We say that a property $P(x)$ depending on $x \in X$
holds *almost everywhere* (abbreviated by *a.e.*) or for *almost all $x \in X$* if
the set of points where it does not hold has measure zero.
{% enddefinition %}

In other words, $P(x)$ a.e. iff
$\mu(\set{x \in X : \neg P(x)}) = 0$.

{% theorem %}
Let $f : X \to \overline{\RR}$ be a nonnegative measurable function. Then

$$
\int_X f \, d\mu = 0
$$

holds if and only if $f$ vanishes almost everywhere.
{% endtheorem %}