blob: a77cf9a8bc965036f6886eca37d33714d755335c (
plain) (
blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
|
---
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 %}
|