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Let $f,g$ be bounded real valued function defined on $[a,b] \subset \mathbb{R}$. My source says that:

Suppose that $f(x)< g(x)$ for all $x\in [a,b]$ and $f,g$ are Riemann Integrable on $[a,b]$.

Then $\int_a^b f \leq \int_a^b g$.

I do not understand why $\le$ is used instead of $<$.

Is this meant to imply that the equality case exists? If so, could you provide an example?

Thanks for your help.

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Seongqjini
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  • Where exactly do you see that statement? – Kenny Lau Aug 02 '16 at 12:30
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    See this question: http://math.stackexchange.com/questions/351157/is-the-riemann-integral-of-a-strictly-positive-function-positive The idea is that if $f\leq g$ and the integrals are equal then the set ${f<g}$ has Lebesgue measure zero. If the inequality is strict everywhere then the integral inequality is also strict. – Beni Bogosel Aug 02 '16 at 12:37
  • @KennyLau In calculus book. – Seongqjini Aug 02 '16 at 12:39
  • @Planche as stated the result is true, if not as precise as possible. Sometimes that is the expense of a simpler proof. Or it could just be sloppy editing. – Umberto P. Aug 02 '16 at 14:19
  • But a simpler proof can be found for a stronger result, namely just assume $f\le g.$ – zhw. Aug 03 '16 at 02:07

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