Proving that the limiting function is in Lp

[bbkrsen585]

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 $$\left\|$$gn$$\left\|$$_p$$\rightarrow$$M < $$\infty$$. How do I show that the pointwise limit g is in Lp?

Thanks.

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.

 

[micromass]

So, you need to show that

$$\int |g|^p<+\infty$$.

Since $$|g_n|^p\rightarrow |g|^p$$, we need to show

$$\int \lim_{n\rightarrow +\infty}{|g_n|^p}<+\infty$$

Now, apply Fatou's lemma...

 

[bbkrsen585]

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.

 

[micromass]

It follows from the continuity of $$|~|^p$$.