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I used following state when I solved some problems.

But I can't prove that state. How can I prove that?

Let $1 $f_k\in L^p$ for $k\ge 1$ such that

  1. There is an $a$ with $\|f_k\|_p \leq a$ for $k=1,2,...$

  2. $\lim f_k(x)=:f(x)$ exists a.$\,$e.

Then $f\in L^p$ (i.$\,$e. $\|f\|_p < \infty$).

  • 0
    What did you try?2012-11-16

1 Answers 1