Measurable function

Remarks

A function $f \colon X \to \R$ is measurable if and only if $\set{x \in X : f(x) > a} \in \Sigma$ for every $a \in \R$. It suffices to check preimages of a family that generates $\mathcal{T}$. Example: 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. Example: the indicator function $\mathbf{1}_A$ is measurable if and only if $A \in \Sigma$. Non-example: if $V$ is a non-measurable subset of $\R$, then $\mathbf{1}_V$ is not Lebesgue measurable, because $\set{\mathbf{1}_V > 1/2} = V$. For Lebesgue measure on $\R$, a measurable function usually means one with $\Sigma$ the Lebesgue measurable sets and $\mathcal{T}$ the Borel sets. A composition of measurable functions is measurable.