I've seen a few other posts about the integral of a positive function, it seems to hinge on it being discontinuous almost nowhere. So what's an example of a discontinuous almost everywhere function that is integrable, positive, and has a zero integral?
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Integrable in what sense? – May 02 '14 at 02:27
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Does "almost nowhere" have a precise meaning? – Umberto P. May 02 '14 at 02:28
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I thought that was the word, discontinuities are measure zero! And @T.Bongers, I'm not sure-it can't hold for Riemann integrable, and I'm not sure why that is-I don't know any other senses, but anyone that gets the job done would work for me... ! – May 02 '14 at 02:30
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"A few other posts" can be found here. – mr_e_man Dec 23 '22 at 03:05
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This doesn't make sense for the Riemann integral, since a function which is any worse than almost-everywhere continuous (that is, the set of discontinuities has positive measure) fails to be Riemann integrable.
For the Lebesgue integral, any measurable function $f : \Bbb R \to \Bbb R$ which is positive on a set of positive measure (i.e. a nonnegative function which is not identified with the zero function) has positive Lebesgue integral. To see this, note that at least one (actually, all but finitely many) of the sets $\{x : f(x) > \frac 1 n\}$ has positive measure; hence $$\int_{\mathbb R} f \ge \int_{\{x : f(x) > \frac 1 n\}} f \ge \frac 1 n m\left\{x : f(x) > \frac 1 n\right\} > 0$$ where $m$ is Lebesgue measure.
So by the two "main" (or at least, arguably the most common) definitions of the integral, any function which is integrable, not negative, and not always zero, has positive integral.