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Is the integral of a strictly positive function on a set of positive measure strictly positive? Thankis a lot
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Yes. Suppose f:A-->R is such a function. We have that
[tex]A=\bigcup_{n\in\mathbb{N}}f^{-1}\left(\left[\frac{1}{n},+\infty\right)\right)[/tex]
and so by subadditivity of the measure
[tex]\mbox{mes}(A)\leq \sum_n\mbox{mes}\left(f^{-1}\left(\left[\frac{1}{n},+\infty\right)\right)\right)[/tex]
Since mes(A)>0, it must be that A_n:=mes(f^{-1}\left(\left[\frac{1}{n},+\infty\right)\right))>0 for some n. By definition
[tex]\int_Af=\sup_h\left{\int_Ah\right}[/tex]
where h:A-->R denotes a positive measurable stair function bounded above by f.
It follows that
[tex]\int_A f\geq \int_{A_n}\frac{1}{n}=\frac{1}{n}\mbox{mes}A_n>0[/tex]