Difference between revisions of "Jets7.3 Old Kiwi" - Rhea

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-Suppose we know the conclusion of problem 8,
-Problem 8
-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>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>
-Now, WTS that <math> f \in L^{p} </math>, which is equivalent to show that <math> |f|^p \in L^{1} </math>
-Let <math> D_n=\{x \in X : |f(x)| \geq  n \}</math>. Then <math> \sum_{n=0}^{\infty}m(D_n)=\sum_{n=0}^{\infty}(n+1)m(E_n)</math>. Thus,
-<math> \sum_{n=0}^{\infty}m(D_n)=\sum_{n=0}^{\infty}(n+1)m(E_n)=\sum_{n=0}^{\infty}m(E_n)+\sum_{n=0}^{\infty}nm(E_n)=m(X)+\sum_{n=0}^{\infty}nm(E_n)</math>.

Revision as of 23:31, 10 July 2008