- #1

Markjdb

- 31

- 0

For integrable [tex] f,g:\left[a,b\right]\rightarrow\mathbb{R} [/tex] with [tex]f(x)\leq g(x)[/tex] for all [tex]x\in\left[a,b\right][/tex], it's a basic property of the riemann integral that

[tex]\[\int_a^b f(x)\,dx \leq \int_a^b g(x)\,dx\][/tex]

My question is whether the strict version of this inequality holds, i.e. if we have the same hypotheses as above, except with [tex]f(x)<g(x)[/tex] for all [tex]x\in\left[a,b\right][/tex], then do we get the following inequality?

[tex]\[\int_a^b f(x)\,dx < \int_a^b g(x)\,dx\][/tex]

This question arose while trying to solve a rather different problem; I feel like it's not true in general, but I haven't yet come up with a counterexample.