# 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. 1.

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

2. 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.

 Title measure Canonical name Measure Date of creation 2013-03-22 11:57:33 Last modified on 2013-03-22 11:57:33 Owner djao (24) Last modified by djao (24) Numerical id 19 Author djao (24) Entry type Definition Classification msc 60A10 Classification msc 28A10 Related topic LpSpace Related topic SigmaFinite Related topic Integral2 Related topic Distribution Related topic LebesgueMeasure Defines measure space Defines probability space Defines probability measure Defines countably additive Defines finitely additive Defines $\sigma$-additive Defines positive measure