Sequence in $L^p(X,M,\mu)$

Question

I have two question.

Suppose that {$f_k$} is a sequence in $L^p(X,M,\mu)$ such that $f(x) = \lim_{k \to \infty} f_k(x)$ exists for $\mu$ -a.e. $x \in X$. Assume $1\le p<\infty$, $\liminf_{k\to \infty} ||f_k||_p = a$ is finite.

First one is proving that $f \in L^p$ and $||f||_p \le a$.

And if additionally assume that $||f||_p = \lim_{k \to \infty} ||f_k||_p $.

Second one is to prove $$\lim_{k \to \infty} ||f-f_k||_p =0 $$

Those are very natural fact, but I want have strict proof of them. How can I approach?

For the second question, see here. – David Mitra Jun 13 '12 at 03:59 — David Mitra, Jun 13 '12 at 03:59

score 4 · Accepted Answer · answered Jun 13 '12 at 03:35

4

1) is easy using Fatou's lemma

answered Jun 13 '12 at 03:35

Robert Israel

448,999

Yeah. It's directly come from Fatou's lemma. thanks. – japee Jun 13 '12 at 04:58

Sequence in $L^p(X,M,\mu)$

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