functions with finite integrals map a measure zero set to infinity

[[concept]]

Topics

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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 $n\in \mathbb{N}$ we have $n{\chi }_F\le f{\chi }_F$ So integrating both sides gives us $nm(F)\le {\int }_{E}f{\chi }_F\le {\int }_{E}f\le \infty$ Thus for all $n$, we have $m(F)\le \frac{1}{n}{\int }_{E}f$. Taking $n\to \infty$, we have $m(F)=0$.

References

References

See Also

Mentions

Mentions

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