[Math] Beppo Levi theorem

Question

In Beppo Levi's theorem, we require that the sequence of measurable functions are $\text{increasing}$. However, does a convergence result for integrals exist which deals with arbitrary sequences of measurable functions $(u_j)$ (as long as they are positive) that are not necessarily increasing, with $u$ being the $\liminf, $\limsup, or even the $\lim$ of these functions?

Or are such results derivable from Beppo Levi? One of these are Fatou's Lemma, but I am less interested in inequalities. Are there others?

Answer 1

There's the Dominated Convergence Theorem, of course:

Let $(f_n)$ be integrable, $\lvert f_n \rvert <g$, $g$ integrable. Then if $f_n \to f$ pointwise a.e., then $$ \lim_n \int f_n = \int f. $$

I assume you know about that, but want something with fewer conditions.

Presumably you're looking for something a bit more like this: there is an extension of Fatou's lemma that tells you when it is an equality. It was discovered by Lieb and Brézis, who call it the missing term in Fatou's lemma:

Let $(f_n) \subset L^p$ be integrable with uniformly bounded integrals, $\int f_n<C$. Suppose also that $f_n \to f$ pointwise a.e. Then $$ \lim_{n} \int \left\lvert \lvert f_n \rvert^p-\lvert f_n-f \rvert^p- \lvert f \rvert^p \right\rvert = 0. $$

You can find the proof of it in Lieb and Loss's Analysis, currently available here. It does require use of the Dominated Convergence Theorem in the proof (which is one reason I mentioned it above), and convexity of $ t \mapsto \lvert t \rvert^p$.