Prove measurability of a function defined from two measurable functions

Question

You have two measurable functions $L$ and $U$ defined on $([0,1],\mathcal{B}[0,1],Leb)$. Define $$ f = \begin{cases} L & \text{if }L=U \\ 0 & \text{otherwise} \end{cases} $$

The text says $f$ is clearly measurable. Its not so obvious to me. Any hints?

Answer 1

If $L$ and $U$ are measurable, for is their difference $L-U$ and hence $$ E=(L-U)^{-1}(\{0\}) $$ is a measurable set, as an inverse of a measurable set. In particular, the characteristic functions of $E$ and $E^c$ are measurable functions.

So $$ f=\chi_E\cdot L, $$ and thus $f$ is measurable as a product of such.