Given a sequence $\{f_n\}$ of measurable functions, why does there exist a subsequence $\{f_{n_k}\}$ such that $\lim_{k \to \infty} \int_E f_{n_k} = \liminf \int_E f_n$? I need to use this in a theorem I am proving, but I don't see how to justify it. Just for your information, I am trying to prove the convergence-in-measure version of Fatou's lemma.
2026-02-23 01:18:31.1771809511
Subsequence of Measurable Functions
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Hint: This is just a statement about real sequences, namely that for a real sequence $(x_n)$, there exists a subsequence $(x_{n_k})$ such that $\lim_{k \to \infty} x_{n_k} = \liminf x_n$. Can you show this instead?