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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
commit | 28407333ffceca9b99fae721c30e8ae146a863da (patch) | |
tree | 67fa2b79d5c48b50d4e394858af79c88c1447e51 /pages/measure-and-integration/measure-theory/signed-measures.md | |
parent | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (diff) | |
download | site-28407333ffceca9b99fae721c30e8ae146a863da.tar.zst |
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-rw-r--r-- | pages/measure-and-integration/measure-theory/signed-measures.md | 33 |
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diff --git a/pages/measure-and-integration/measure-theory/signed-measures.md b/pages/measure-and-integration/measure-theory/signed-measures.md new file mode 100644 index 0000000..77b2416 --- /dev/null +++ b/pages/measure-and-integration/measure-theory/signed-measures.md @@ -0,0 +1,33 @@ +--- +title: Signed Measures +parent: Measure Theory +grand_parent: Measure and Integration +nav_order: 10 +--- + +# {{ page.title }} + +{% definition Signed Measure %} +A *signed measure* on a σ-algebra $\mathcal{A}$ on a set $X$ +is a mapping $\mu : \mathcal{A} \to [-\infty,\infty]$ such that +{: .mb-0 } + +- $\mu(\varnothing) = 0$, +- either there is no $A \in \mathcal{A}$ with $\mu(A) = -\infty$ + or there is no $A \in \mathcal{A}$ with $\mu(A) = \infty$, +- for every sequence $(A_n)_{n \in \NN}$ of + pairwise disjoint sets $A_n \in \mathcal{A}$ + {: .my-0 } + + $$ + \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen + = \sum_{n=0}^{\infty} \mu(A_n). + $$ +{% 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 %} |