I have the following task from my textbook: Let { f <

Emanuel Keith

Emanuel Keith

Answered question

2022-06-22

I have the following task from my textbook:
Let { f n } n N be a sequence of measurable function on M with
f n f  a.s., 
where f is also a measurable function. Here, a.s. means almost surely (= almost everywhere).
Show: if there exists a nonnegative measurable function g satisfying the following conditions:
| f n | g  a.s. for all  n N
and
M g p d μ <  for one  p > 0 ,
then
M | f n f | p d μ 0  for  n .
These notations remind me a little bit of the Hölder inequality.
What I have got:
I thought that we have
| f n | p g p a . s .
Thus
M | f n | p d μ M g p d μ i n f t y
And as
f n f
then the rest follows?

Answer & Explanation

lisicw2

lisicw2

Beginner2022-06-23Added 11 answers

For p > 1 ,, the function x x p is convex so
| f n f | p = 2 p | f n 2 + ( f ) 2 | p 2 p 1 ( | f n | p + | f | p )
so
2 p 1 ( | f n | p + | f | p ) | f n f | p 0.
In the standard way that dominated convergence theorem is proven, just apply Fatou's lemma.

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