Measurable function: Difference between revisions
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In [[mathematics]], a [[function]] ''f'' that maps each element of a [[measurable space]] <math>(X,\mathcal{F}_X)</math> to an element of another measurable space <math>(Y,\mathcal{F}_Y)</math> is said to be '''measurable''' (with respect to the [[sigma algebra]] <math>\mathcal{F}_X</math>) if for any set <math>A \in \mathcal{F}_Y</math> it holds that <math>f^{-1}(A) \in \mathcal{F}_X</math>, where <math>f^{-1}(A)=\{x \in X \mid f(x) \in A\}</math>. | In [[mathematics]], a [[function]] ''f'' that maps each element of a [[measurable space]] <math>\scriptstyle (X,\mathcal{F}_X)</math> to an element of another measurable space <math>\scriptstyle (Y,\mathcal{F}_Y)</math> is said to be '''measurable''' (with respect to the [[sigma algebra]] <math>\scriptstyle \mathcal{F}_X</math>) if for any set <math>\scriptstyle A \in \mathcal{F}_Y</math> it holds that <math>\scriptstyle f^{-1}(A) \in \mathcal{F}_X</math>, where <math>\scriptstyle f^{-1}(A)=\{x \in X \mid f(x) \in A\}</math>. | ||
Latest revision as of 05:55, 12 May 2008
In mathematics, a function f that maps each element of a measurable space to an element of another measurable space is said to be measurable (with respect to the sigma algebra ) if for any set it holds that , where .
