According to Lusin's theorem (and the slightly weaker converse of that result), measurable functions on locally compact topological spaces that are equipped with a Borel regular measures may be characterized as follows:

Definition. Let $X$ be a locally compact space and $\mu$ be a regular measure on $X$. We call $f:X\to\mathbb C$ measurable if for every $\varepsilon>0$ and for every compact set $K\subset X$, there exists a compact set $L\subset X$ such that $\mu(K\setminus L)<\varepsilon$, and $f$ restricted to $L$ is continuous.

In fact, Bourbaki defines a measurable functions as such in their volume on integration. To me, this definition of a measurable function seems much less intuitive than the one I am used to, where we require that the preimage of every measurable set be measurable. My question is therefore the following: Are there any situations in which this definition of measurability is clearly advantageous or easier to work with?