Background
I had a very good and bright student with some decent exposure to markets and the standard college math curriculum who got overwhelmed during a mathematical finance course due to unnecessary off-topic formalisms. In particular, the curriculum of the class was a bit misleading because it promised to follow a fairly practical book (Tomas Bjork's Arbitrage Theory in Continuous Time) as its main text, while the actual class went far beyond that, largely ignoring the text. The course spent a considerable amount of time dealing with concepts such as Radon-Nikodym derivatives, Formal Measure Theory, Filtrations, Martingales (just to name a few) which - crucially - the book only mentions as an addendum in the appendix.
Of course, there is absolutely nothing wrong with any of the topics above. It is just that my student was expecting a largely more applied class in Finance, versus a purely theoretical one in Measure Theory and Martingales.
Questions
This leads me to some questions.
What is the extent to which abstract topics in mathematical finance, such as Radon-Nikodym derivaties are actually relevant to markets in the real world? Do finance instructors spend any time in the markets or do they prefer to engage in a series of endless mental gymnastics often for their own sake? (Incidentally, I am not the first one asking these questions)
Is this top-down approach and over-formalism necessary? Does one really need to know Radon-Nikodym derivatives (and all their intricate details) in an introductory Math of Finance class? Especially one without any prerequisites which uses a largely informal book as its main text and which is not designed for math majors?
EDITS
Post shortened, following the suggestions of @Noah Schweber (Thank you!)
There is also a link to a chat below (thank you @roddik!). Thank you all for being constructive and respectful.