I always had the dream to design a course for my graduate students like "mathematical models of the continuum". This course should cover history of real numbers, the Measure Problem, the Banach-Tarski Paradox, but also Baire/Cantor/Polish spaces, and it should look at infinite games and determinacy as well.

Obviously I am conscious about the text books and articles covering all these concepts. However, I seem to be unable to find a rigorours, graduate-level monograph that captures all of this in one, which I could potentially use as a text.

Would someone be able to point me to a good reference for that?

descriptive set theory, which treats many of the topics you seem to have in mind. $\endgroup$