| Line 8: | Line 8: | ||
<math>f \in L^1</math> if and only if <math>\sum_{n=1}^{\infty}nm(E_n) < \infty.</math> | <math>f \in L^1</math> if and only if <math>\sum_{n=1}^{\infty}nm(E_n) < \infty.</math> | ||
| − | First, if <math> m(X)= | + | First, if <math> m(X)= \infty </math>, it's done. Hence let's suppose that <math> m(X)<\infty </math> |
| + | |||
| + | Now, WTS that <math> f \in L^{p} </math>. | ||
Revision as of 22:59, 10 July 2008
Suppose we know the conclusion of problem 8,
Problem 8 Let $ X $ be a finite measure space. If $ f $ is measurable, let
$ E_n = \{x \in X : n-1 \leq |f(x)| < n \} $. Then
$ f \in L^1 $ if and only if $ \sum_{n=1}^{\infty}nm(E_n) < \infty. $
First, if $ m(X)= \infty $, it's done. Hence let's suppose that $ m(X)<\infty $
Now, WTS that $ f \in L^{p} $.
