$ \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\N}{\mathbb{N}} \newcommand{\R}{\mathbb{R}} $The Problem: Let $\set{f_k}_{k \in \N}$ be a sequence of functions such that
- $f_k : E \subseteq \R^n \to \R_{\ge 0} \cup \set \infty$ (i.e. each is nonnegative, possibly infinite)
- $f_k$ is measurable
- $f_k \to f$ pointwise a.e. on $E$
- $f_k \le f$ a.e. on $E$
(where the measure-based properties are w.r.t. to the Lebesgue measure $\mu$). Is it true that
$$\lim_{k \to \infty} \int_E f_k \, d \mu = \int_E f \, d \mu?$$
My Understanding/Attempts:
My running guess is that it's not true, but I'm struggling to find a counterexample. This bears a lot of resemblance and relation to the monotone convergence theorem, dominated convergence theorem, and Fatou's lemma, particularly in how it deals with switching limits and integrals -- however I'm struggling to find one of the usual examples that means all of the criteria (in particular that $f_k \le f$).
I've searched through a number of such threads on MSE and a few other sites and can't quite find a post that uses these criteria, even if they're close:
For instance, if we have monotonicity in the $f_k$, so $f_1 \le f_2 \le \cdots$, then equality holds as I understand it. So perhaps we need something that oscillates in some way works?
Using infinite-valued functions might also work, from what I've looked up, but I'm less sure.
Dominated convergence uses a dominating function $g$ that need not be the limit $f$; that's possibly useful somehow?
Can anyone give me an idea as to where I should go to look for such a counterexample? Whether it be the function outright, or just the gist of what I should be trying to do in order to create it (handy tricks, ideas, whatnot). Or, of course, correcting me and telling me that I'm wrong and that the claim is actually true -- either one is appreciated.
Hints: i) If $\int f <\infty,$ try DCT.
ii) If $\int f=\infty,$ try Fatou's Lemma