Proving that the limiting function is in Lp

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  • #1

Homework Statement

Hi guys. I have one question regarding convergence in Lp. Suppose gn converges to g mu-almost everywhere on [0,1]. Suppose further that [tex]\left\|[/tex]gn[tex]\left\|[/tex]p[tex]\rightarrow[/tex]M < [tex]\infty[/tex]. How do I show that the pointwise limit g is in Lp?


Homework Equations

So far, I know this is not true if we only know that gn goes to g mu-almost everywhere. I just don't see how the additional condition that the Lp-norms converge to some finite number implies that the limiting function g is also in Lp.

Answers and Replies

  • #2
So, you need to show that

[tex]\int |g|^p<+\infty[/tex].

Since [tex]|g_n|^p\rightarrow |g|^p[/tex], we need to show

[tex]\int \lim_{n\rightarrow +\infty}{|g_n|^p}<+\infty[/tex]

Now, apply Fatou's lemma...
  • #3
But, how do we know that |gn|^p converges to |g|^p a.e.?

If gn goes to g almost everywhere does that imply that |gn|^p converges to |g|^p a.e.? That's not so clear to me. Thanks.
  • #4
It follows from the continuity of [tex]|~|^p[/tex].

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