I just had a question regarding an exercise from Tao's Intro to Measure Theory (Exercise 1.4.46). The question is as follows:
Let $(X, B, \mu)$ be a measure space, and let $f_1, f_2, …: X \to C$ be a sequence of measurable functions that converge pointwise $\mu-a.e.$ to a measurable limit $f: X \to C$. Suppose that there is an unsigned absolutely integrable function $G, g_1, g_2, … : X \to [0, +\infty]$ such that $|f_n|$ are pointwise a.e. bounded by $G + g_n$, and that $\int_X g_n d\mu \to 0$ as $n \to \infty$. Show that $\lim_{n\to\infty} \int_X f_n d\mu \to \int_X f d\mu$.
I've seen the generalized Dominated Convergence Theorem before, but I'm having some serious trouble trying to figure out how to proceed, since the convergence of the g_n integrals to zero only implies the convergence of g_n in measure, and doesn't directly imply anything regarding the pointwise convergence of g_n (I'm thinking about the typewriter function). My current direction is that I'd like to find something out regarding the pointwise convergence of the g_n's so that I appropriately use Fatou's lemma.
So I suppose my questions are, is there a way to show that the g_n's converge pointwise to 0? Considering the assumptions in the question, I feel like any sequence of functions bounded by another sequence of functions whose integrals converge to zero, but do not converge pointwise, cannot itself converge pointwise.
If this is not the right direction, .. is convergence in measure supposed to be enough?
If both of these are not the right direction, please provide a small hint as to the correct direction… thanks!
Best Answer
To prove that $\int f_n \to \int f$ it is enough to prove that every subsequence of $(f_n)$ has a further subsequence for which the limit relation holds.
Also, convergence in measure implies a.e. convergence for a subsequnce. Combine these two facts with the result below (with $G+g_n$ in place of $g_n$) to finish the proof:
General Lebesgue Dominated Convergence Theorem