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  "body": "\n## Definition\n\nLet $(X, \\Sigma)$ and $(Y, \\mathcal{T})$ be [[measurable spaces|N4KDT83V3S]].\nA [[function|0EPAT51DXQ]] $f \\colon X \\to Y$ is *measurable* if $f^{-1}(B) \\in \\Sigma$ for every $B \\in \\mathcal{T}$.\nA function $f \\colon X \\to \\R$ is measurable if it is measurable for the [[Borel sigma-algebra|0WQXY32Q8D]] on $\\R$.\n\n## Remarks\n\nA function $f \\colon X \\to \\R$ is measurable if and only if $\\set{x \\in X : f(x) \u003e a} \\in \\Sigma$ for every $a \\in \\R$.\nIt suffices to check preimages of a family that generates $\\mathcal{T}$.\nExample: every continuous $f \\colon \\R \\to \\R$ is Borel measurable, since preimages of open sets are open and the open sets generate the Borel sets.\nExample: the indicator function $\\mathbf{1}_A$ is measurable if and only if $A \\in \\Sigma$.\nNon-example: if $V$ is a non-measurable subset of $\\R$, then $\\mathbf{1}_V$ is not Lebesgue measurable, because $\\set{\\mathbf{1}_V \u003e 1/2} = V$.\nFor Lebesgue measure on $\\R$, a measurable function usually means one with $\\Sigma$ the Lebesgue measurable sets and $\\mathcal{T}$ the Borel sets.\nA composition of measurable functions is measurable.\n",
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