Is the "continuous on compact subsets" characterization of measurable functions actually useful? 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 regular measure 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?
 A: Tautological answer: the cases where you know the measure of compact sets (as in section 3 of chapter IX of Bourbaki), but not the class of all measurable sets (which you then can define using the above definition applied to the indicator function of the set). So the next question is: should one take the topology (or even only the compact subsets), and not the class of all measurable set, as starting point? Answer: perhaps. To understand the answer, one can look at the review of Bourbaki's integration by G.A.Edgar, pag. 82-84 of Mathematical intelligencer, 1981 (I have no month in my copy of that review). It is worth reading; among other things one reads: 
There are valid motives for doing measure theory using Radon measures instead of abstract measures. An important one is the problem of disintegration of measures (or of "regular conditional probabilities"). [...] Another problem with abstract measures concerns the connection between point-maps and set-maps. [...] A third shortcoming of abstract measures is called the "image-measure catastrophe" by L.Schwartz [in his book "Radon measures on arbitrary topological spaces and cylindrical measures"]. [...] On the other hand, there are also many reasons to use abstract measures instead of Radon measures.
For these reasons, he cites the preceding reviews of Bourbaki's integration by Halmos and by Munroe in the bull. AMS: 59 (1953) and 65 (1958); you can find them easily on line. He cites another reason, the "barycenter catastrophe".
Even A.Weil himself was not completely satisfied with Bourbaki's integration (but for reasons different from the requester); the see Weil's comments in the third volume of his collected papers, where he refers to a compact system of sets as starting point and Schwartz's book for some more details. I only remark that Weil's starting point would provide a connection with one concept categorical topologist studied around 1970: compactly generated (weak) Hausdorff spaces as convenient setting for topology (convenient meaning that the exponential law is available)
