1 files changed, 27 insertions, 0 deletions
diff --git a/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md b/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md
new file mode 100644
index 0000000..a77cf9a
--- /dev/null
+++ b/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md
@@ -0,0 +1,27 @@
+---
+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 %}