pages/measure-and-integration/measure-theory/measures.md


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---
title: Measures
parent: Measure Theory
grand_parent: Measure and Integration
nav_order: 4
---

# {{ page.title }}

{% definition %}
A *measure* on a σ-algebra $\mathcal{A}$ on a set $X$
is mapping $\mu : \mathcal{A} \to [0,\infty]$ such that

- $\mu(\varnothing) = 0$,
- for every sequence $(A_n)_{n \in \NN}$ of
  pairwise disjoint sets $A_n \in \mathcal{A}$

  $$
  \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen
  = \sum_{n=0}^{\infty} \mu(A_n).
  $$
{% enddefinition %}

{% definition Measure Space %}
A *measure space* is a triple $(X,\mathcal{A},\mu)$ of
a set $X$,
a σ-algebra $\mathcal{A}$ on $X$
and a measure $\mu$ on $\mathcal{A}$.
{% enddefinition %}