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diff --git a/pages/measure-and-integration/measure-theory/measures.md b/pages/measure-and-integration/measure-theory/measures.md new file mode 100644 index 0000000..637ab0c --- /dev/null +++ b/pages/measure-and-integration/measure-theory/measures.md @@ -0,0 +1,29 @@ +--- +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 %} |