Integration Inequality: f(x) vs g(x)

[JG89]

I know that if $$\forall x \in E \subset \mathbb{R}^n$$ we have $$f(x) \le g(x)$$ then it is true that $$\int_E f \le \int_E g$$.

However, is it also true that if $$\forall x \in E$$ we have $$f(x) < g(x)$$ then $$\int_E f < \int_E g$$?

 

[Stephen Tashi]

Did you mean to also specifiy that $E$ is a set with non-zero measure? i.e.
$$\int_E 1 > 0$$ ?

 

[JG89]

Yeah. I can see how without giving that extra stipulation what I am asking isn't true.

 

[micromass]

And you probably also want that the integral isn't infinity... In those cases, I think it's true what you're saying...