TITLE: Why isn't Dominated Convergence Theorem taught in intro analysis
QUESTION [23 upvotes]: In a course based off a book like Rudin's Principles of Mathematical Analysis that does non-measure theoretic analysis why isn't dominated convergence taught? It would be useful since continuous functions are Lebesgue measurable. Is there not a way to prove a non-measure theoretic version of this statement?

REPLY [3 votes]: I would wager there were well known "dominated convergent theorems" around before the Lebesgue revolution. Two examples come to mind - one, the case of infinite series; this DCT is quite simple to prove, no measure theory needed. The second one concerns integration on an interval. We might for example have a sequence of continuous functions $f_n$ on $(0,\infty)$ that are bounded in absolute value by some $g:(0,\infty) \to [0,\infty).$ We further assume that $\int_0^\infty g <\infty,$ and that $f_n\to f$ uniformly on $[a,b]$ for any $0<a<b<\infty.$ In this case we get $\lim \int_0^\infty f_n \to \int_0^\infty f.$ This is also not hard to prove, and again, no measure theory is needed.
I think it would be good to cover these "easy" DCTs in a Rudin level first course in analysis. Then when students see the big DCT in measure theory, they are ready for it.