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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?

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    For the second question, see [here](http://math.stackexchange.com/questions/51502/pointwise-and-convergence-of-lp-norms-implying-convergence-in-lp).2012-06-13

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1) is easy using Fatou's lemma

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    Yeah. It's directly come from Fatou's lemma. thanks.2012-06-13