Correct on your first point. I do not mean I learned the specific version published by Bourbaki, but rather in the modern mathematical style (hence the "a la", as opposed to 19th century hand waving style that nearly all undergraduate first-courses do).

And by the modern style, I meant starting out with analysis, defining the axioms of what measures are, demonstrating the existence of nonmeasurable sets with the axiom of choice, etc. IIRC, the course did rely on Borel algebras for its buildup, but did not openly buildup from sigma algebra machinery.