PlanetMath: measurable function

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measurable function

(Definition)

Let $\big(X,\mathcal{B}(X)\big)$ and $\big(Y, \mathcal{B}(Y)\big)$ be two measurable spaces. Then a function $f\colon X\to Y$ is called a measurable function if:

$\displaystyle f^{-1}\big(\mathcal{B}(Y)\big) \subseteq \mathcal{B}(X)$

where $f^{-1}\big(\mathcal{B}(Y)\big) = \{f^{-1}(E)\mid E\in\mathcal{B}(Y)\}$ .

In other words, the inverse image of every $\mathcal{B}(Y)$ -measurable set is $\mathcal{B}(X)$ -measurable. The space of all measurable functions $f\colon X\to Y$ is denoted as

$\displaystyle \mathcal{M}\big(\big(X,\mathcal{B}(X)\big),\big(Y, \mathcal{B}(Y)\big)\big).$

Any measurable function into $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ , where $\mathcal{B}(\mathbb{R})$ is the Borel sigma algebra of the real numbers $\mathbb{R}$ , is called a Borel measurable function.¹ The space of all Borel measurable functions from a measurable space $(X,\mathcal{B}(X))$ is denoted by $\displaystyle{\mathcal{L}^0\big(X,\mathcal{B}(X)\big)}$ .

Similarly, we write $\displaystyle{\bar{\mathcal{L}}^0\big(X,\mathcal{B}(X)\big)}$ for $\displaystyle{\mathcal{M}\big(\big(X,\mathcal{B}(X)), (\bar{\mathbb{R}},\mathcal{B}(\bar{\mathbb{R}})\big)\big)}$ , where $\mathcal{B}(\bar{\mathbb{R}})$ is the Borel sigma algebra of $\bar{\mathbb{R}}$ , the set of extended real numbers.

Remark. If $f:X\to Y$ and $g:Y\to Z$ are measurable functions, then so is $g\circ f:X\to Z$ , for if is $\mathcal{B}(Z)$ -measurable, then $g^{-1}(E)$ is $\mathcal{B}(Y)$ -measurable, and $f^{-1}\big(g^{-1}(E)\big)$ is $\mathcal{B}(X)$ -measurable. But $f^{-1}\big(g^{-1}(E)\big)=(g\circ f)^{-1}(E)$ , which implies that $g\circ f$ is a measurable function.

Example:

Let be a subset of a measurable space then the characteristic function $\chi_E$ is a measurable function if and only if is measurable.

Footnotes

...¹: More generally, a measurable function is called Borel measurable if the range space is a topological space with $\mathcal{B}(Y)$ the sigma algebra generated by all open sets of .

"measurable function" is owned by CWoo. [ full author list (3) | owner history (2) ]

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Other names:

Borel measurable

Also defines:

Borel measurable function

Keywords:

measurable

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Cross-references: measurable, characteristic function, subset, implies, extended real numbers, open sets, generated by, sigma algebra, topological space, range, real numbers, Borel sigma algebra, inverse image, function, measurable spaces
There are 47 references to this entry.

This is version 14 of measurable function, born on 2002-07-21, modified 2006-12-09.
Object id is 3176, canonical name is MeasurableFunctions.
Accessed 14576 times total.

Classification:

AMS MSC:

28A20 (Measure and integration :: Classical measure theory :: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence)

Pending Errata and Addenda

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