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Let ($X,\Sigma, \mu)$ be a measure space. Suppose that $\mu$ is semifinite, $E \in \Sigma$, and $\mu(E)=\infty$. Prove that if $C>0$, then there exists some measurable set $A \subseteq E$ that satisfies $C < \mu(A) < \infty.$

I have a hint:

Define $C =$sup{ $\mu(F)$ $:$ $F \in \Sigma, F\subseteq E, \mu(F) < \infty$}. If $C<\infty$, then there exists measurable sets $F_k \subseteq E$ with finite measure such that $\mu(F_k)\to C$.

A quick definition: If every set $E \in \Sigma$ with $\mu(E)= \infty$ contains an $F \subseteq E$ such that $F \in \Sigma$ and $\mu(F) < \infty$, then $\mu$ is semifinite.

I know the first step is to find a contradiction if C is finite in the hint.

HCS
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    I'm confused as to what the hint is saying: as I have understood, you can't just define $C$ but rather must prove that given $C$, we have $A \subset E$ with $C < \mu(A) < \infty$, no? – qualcuno Feb 27 '19 at 03:37
  • https://math.stackexchange.com/questions/234930/if-a-measure-is-semifinite-then-there-are-sets-of-arbitrarily-large-but-finite –  Feb 27 '19 at 03:38
  • Thanks for the links. Yeah the hint is what confused me. – HCS Feb 27 '19 at 04:07

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