functions with finite integrals map a measure zero set to infinity

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Theorem

Let $f \in L^{+} (E)$ and let $\int_{E} f < \infty$ . Then the set

F = {x \in E : f (x) = \infty}

must have measure 0.

Proof

We know for all $n \in N$ we have

n χ_{F} \leq f χ_{F}

So integrating both sides gives us

n m (F) \leq \int_{E} f χ_{F} \leq \int_{E} f \leq \infty

Thus for all $n$ , we have $m (F) \leq \frac{1}{n} \int_{E} f$ . Taking $n \to \infty$ , we have $m (F) = 0$ .

functions with finite integrals map a measure zero set to infinity

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