= Lebesgue measurable function
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A <Lebesgue measurable function> $f:\mathbb R\to\mathbb R$ has a <Lebesgue measurable set> as the <preimage> of every open <subset> of $\mathbb R$. Equivalently, use every <Borel set> in the target: the sets whose preimages are measurable form a <sigma-algebra> containing all open sets. The source carries the <Lebesgue sigma-algebra> and the target its <Borel sigma-algebra>. A <Borel measurable function> on $\mathbb R$ instead requires the preimages to be Borel in the source as well. Restricting the source to a <Lebesgue measurable set> uses its induced sigma-algebra.
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