it is only interesting to take integrals of functions with positive measure

[[concept]]

Topics

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Theorem

If $E\subset \mathbb{R}$ is a set with $m(E)=0$, then for all $f\in L^{+}(E)$ we have ${\int }_{E}f=0$

ie it is only interesting to take integrals over functions with positive measure

Proof

From the definition, we have $f\in L^{+}(E)$. If $\phi$ is a simple function such that $\phi ={\sum }_{i=1}^n{a}_i{\chi }_{A_i}$ and $\phi \le f$, then $m(A_i)\le m(A)=0$. So in the sum, all terms must be $0$. Thus we always have ${\int }_E\phi =0$ and the supremum over all such $\phi$ is also $0$.

References

References

See Also

Mentions

Mentions

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