What is measurable function?

Let $(X, \mathcal A)$, $(Y, \mathcal B)$ be measurable spaces. $f \colon X \to Y$ is measurable if $f^{-1}(B) \in \mathcal A$ for every $B \in \mathcal B$. For real (or $[-\infty,+\infty]$-valued) functions, $Y = \R$ carries its Borel $\sigma$-algebra, and it suffices to check $f^{-1}(\intoo t{+\infty}) = \{f > t\} \in \mathcal A$ for all $t \in \R$: the good sets $\{B : f^{-1}(B) \in \mathcal A\}$ form a $\sigma$-algebra (preimages commute with set operations) containing the generating rays (Definition 9.2, Method 9.17).