PlanetMath: measure

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measure

(Definition)

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:

$\mu(A) \geq 0$ for $A \in \mathcal{B}(E)$ , with equality if $A = \emptyset$
$\mu(\bigcup_{i=0}^\infty A_i) = \sum_{i=0}^\infty \mu(A_i)$ 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.

"measure" is owned by djao. [ full author list (2) ]

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See Also:

-space, $\sigma$ -finite, Lebesgue integral, probability distribution function, Lebesgue measure

Also defines:

measure space, probability space, probability measure, countably additive, finitely additive, $\sigma$ -additive, positive measure

Attachments:: properties for measure (Theorem) by matte
Radon measure (Definition) by ptr
Borel measure (Definition) by asteroid

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Cross-references: Lebesgue measure, unions, finite, algebra, countable additivity, property, reals, restricted, term, pairwise disjoint, sequence, equality, extended real numbers, function, measurable space
There are 258 references to this entry.

This is version 15 of measure, born on 2001-11-11, modified 2008-11-15.
Object id is 756, canonical name is Measure.
Accessed 45155 times total.

Classification:

AMS MSC:	28A10 (Measure and integration :: Classical measure theory :: Real- or complex-valued set functions)
	60A10 (Probability theory and stochastic processes :: Foundations of probability theory :: Probabilistic measure theory)

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