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Homemeasure

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# measure

Let $(E,\mathcal{B}(E))$ be a measurable space. A *measure* on $(E,\mathcal{B}(E))$ is a function $\mu\colon\mathcal{B}(E)\to\mathbb{R}\cup\{\infty\}$ with values in the extended real numbers such that:

1. $\mu(A)\geq 0$ for $A\in\mathcal{B}(E)$, with equality if $A=\emptyset$

2. $\mu(\bigcup_{{i=0}}^{\infty}A_{i})=\sum_{{i=0}}^{\infty}\mu(A_{i})$ for any sequence of pairwise disjoint sets $A_{i}\in\mathcal{B}(E)$.

Occasionally, the term *positive measure* is used to distinguish measures as defined here from more general notions of measure which are not necessarily restricted to the non-negative extended reals.

The second property above is called countable additivity, or $\sigma$-additivity. A *finitely additive measure* $\mu$ has the same definition except that $\mathcal{B}(E)$ is only required to be an algebra and the second property above is only required to hold for finite unions. Note the slight abuse of terminology: a finitely additive measure is not necessarily a measure.

The triple $(E,\mathcal{B}(E),\mu)$ is called a *measure space*. If $\mu(E)=1$, then it is called a *probability space*, and the measure $\mu$ is called a *probability measure*.

Lebesgue measure on $\mathbb{R}^{n}$ is one important example of a measure.

## Mathematics Subject Classification

60A10*no label found*28A10

*no label found*

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