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Problem 8 | Problem 8 | ||
| − | Let <math>X</math> be a finite measure space. If <math>f</math> is measurable, let | + | Let <math> X </math> be a finite measure space. If <math> f </math> is measurable, let |
<math>E_n = \{x \in X : n-1 \leq |f(x)| < n \}</math>. Then | <math>E_n = \{x \in X : n-1 \leq |f(x)| < n \}</math>. Then | ||
| 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)=/infty </math>, it's done. Hence let's suppose that <math> m(X)<\infty </math> | + | First, if <math> m(X)= /infty </math>, it's done. Hence let's suppose that <math> m(X)<\infty </math> |
Revision as of 22:56, 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 $
