I have never been asked by a quant "Tell me measure theory".
It's not computable/constructive.
One major exception is Radon-Nikodym parameter estimation with MLE as discussed on other forum.
you are missing the purpose of measure theory. some people think as follows: if math topic X is not constructive, it is useless for application, therefore it is pointless to learn it. of course, this is naive and stupid. one only needs to look at Hardy's comments about number theory as evidence (constructive proof!) of this stupidity. sometimes even i think like that: "this is not constructive". and then i think of Hardy.
if we use your logic, you would have to present a constructive proof of the real numbers. why? because quants use continuous time models, i.e. [0,1]. how do you know that [0,1] exists? actually you dont. its obvious, right? if you are set on being constructive in mathematics, you will be very miserable. i am not even going to get into constructing the wiener process on [0,1]. guess what though - all proofs of the existence of [0,1] and the wiener process on [0,1] use measure theory (and set theory) at some point. yet we use these objects every day and are often asked about them. do quants ask you to prove the existence of the wiener process? no. do they ask you about properties of the wiener process? yes. usually they ask because the properties can be exploited by a computer. but the properties follow from the definition. you would have to be a downright fool to look at the properties but not the definition, the construction, etc, of the wiener process. and all of that 'deeper' thinking is usually ticked off in measure theory / functional analysis / stochastic analysis courses.
the same holds in mathematical finance. guess what, nobody uses the Black Scholes model anymore - quants use the Black Scholes pricing equation to recover derivative prices given an implied volatility parameter, and also to hedge (functional derivatives of pricing equation). how do we know there is an underlying process that replicates that derivative? unless we impose extreme constraints (which dont hold in reality), we don't. so we are not really being constructive when modelling, either. maybe you can call it pseudo-constructive - because you can perform some MLE estimation and find an 'optimal' value of a parameter, you call this constructive. but the RND arises from some underlying process, and we dont even know if that process replicates the derivative. so why is that estimate useful? It might not be. often it is totally useless. this kind of knowledge can help when completing modelling work as it gives you theoretical perspective.
actually, quants are interested in measure theory, but it is not so obvious. many quant interview questions (for example, coin tossing experiment given some a priori information) are essentially questions on common sense + integrals + filtrations. they aren't written down like that on paper - but knowledge of those topics helps. derivatives that have forward skew/smile, like cliquets, etc, require functional derivatives (measure theory & functional analysis). generally any pay off with convexity will require measure theory at some point, for example using Dominated Convergence theorems to prove that something is submartingale. but this isn't obvious. and its certainly not written down.
and even more generally, the Daniell integral is the best example to understand why measure theory is important. there was a massive rush to present integration following Daniell's approach. guess what the clever mathematicians realised? on a simple level, the Daniell integral is easier to construct than the Lebesgue integral. but any analysis will require proving the same theorems that you do for the Lebesgue integral in measure theory just now for the Daniell integral. And to prove these theorems is a nightmare, because you are no longer working with measures. nor is intuitive. so guess what happened? people accepted measure theory. it definitely has a problem with being constructive, but nothing else is better. it is all that we have. after the Riemann integral of course, which in theory is completely useless but in practice is all that is used.