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

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Theorem

If $E \subset R$ 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 $φ$ is a simple function such that $φ = \sum_{i = 1}^{n} a_{i} χ_{A_{i}}$ and $φ \leq f$ , then $m (A_{i}) \leq m (A) = 0$ . So in the sum, all terms must be $0$ . Thus we always have $\int_{E} φ = 0$ and the supremum over all such $φ$ is also $0$ .

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

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