Measurable function

A measurable function between measurable spaces $(X,\Sigma)$ and $(Y,\mathcal T)$ is a function $f:X\to Y$ such that for every $B\in \mathcal T$, the preimage $f^{-1}(B)$ lies in $\Sigma$.

Measurability depends on the choice of sigma-algebras on domain and codomain; in particular, using the Borel sigma-algebra connects measurability to topology. For example, a continuous map between topological spaces is Borel measurable.

Examples:

• If $f:\mathbb R\to\mathbb R$ is continuous (in the usual topology), then $f$ is measurable as a map $(\mathbb R,\mathcal B(\mathbb R))\to(\mathbb R,\mathcal B(\mathbb R))$.
• If $A$ is a measurable set in $(X,\Sigma)$, then its indicator function $\mathbf 1_A:X\to\{0,1\}$ is measurable (with $\{0,1\}$ carrying its power-set sigma-algebra).
• Every simple function is measurable by definition.