Let $(X, \mathcal{X}, \mu)$ be a measure space. Suppose $(f_n)_{n\in\mathbb{N}} \subset L^1(\mu)$ is such that:
- $f_n \to f \in L^1(\mu)$ a.e.
- $ \int_A | f_n | \ \mathrm{d} \mu \to \int_A |f| \ \mathrm{d} \mu$
Then, is it true that $\int_A |f_n - f| \ \mathrm{d}\mu \to 0$?
I have managed to prove the claim under the additional assumption that $\mu(A) < \infty$, using Egorov's Theorem and absolute continuity of the Lebesgue integral. However, is it still true if:
- $\mu(A) = \infty$ and $A$ is $\sigma$-finite? The difficulty here is that (2) does not necessarily hold over the finite measure subsets which cover $A$.
- $\mu(A) = \infty$ and $A$ is not $\sigma$-finite?
Proofs/counter-examples for the two cases above are welcome.
