Is there a function $f \gt 0$ such that $\int f dx=0$?

Question

If $$f \ge 0$$, then $$\int f dx \ge 0$$.

There is a function such that

$$f \gt 0$$ but $$\int f dx=0$$?

Then such function doesn't exist. You can find a proof below. You can also argue it by using the fact $f$ is Riemann integrable $\iff$ it's bounded and continuous almost everywhere. — John, Oct 29 '14 at 11:32
Thank you for commet. Can I do this without measure theory?? — user148928, Oct 29 '14 at 14:04

score 1 · Answer 1 · edited Apr 13 '17 at 12:19

1

If you choose your domain of integration be a measure zero set, then any Lebesgue integrable (in particular Riemann integrable) function has zero integral.

If you consider Riemann integrable function on $[a,b]$ with $a<b$, then it's not possible, see a proof here.

edited Apr 13 '17 at 12:19

Community

1

answered Oct 29 '14 at 06:26

John

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Is there a function $f \gt 0$ such that $\int f dx=0$?

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