Gatech Math 6701 [patched]

This is exactly the Dominated Convergence Theorem. The proof requires:

The primary architect of this transformation is the Lebesgue integral. While the Riemann integral suffices for continuous functions and nice domains, it collapses under the weight of more pathological examples, such as the Dirichlet function (which is 1 on rationals and 0 on irrationals). MATH 6701 opens by exposing the Riemann integral’s limitations, establishing the need for a more powerful and flexible theory. The course then proceeds through a meticulously structured sequence: first, the definition of a (\sigma)-algebra and the concept of a measurable set; second, the construction of a measure (starting with Lebesgue measure on (\mathbbR^n)); third, the definition of measurable functions; and finally, the construction of the Lebesgue integral via limits of simple functions. Each step is a logical fortress, built upon the last, requiring students to internalize abstract definitions and deploy them in proofs of foundational theorems like the Monotone Convergence Theorem, Fatou’s Lemma, and the Dominated Convergence Theorem. gatech math 6701

When you are lost in a proof, look at the map to see the logical dependencies. This is exactly the Dominated Convergence Theorem