pr.probability - Constraints for different probability measures to have the same expectation.

Take different $D_i in mathbb{R} rightarrow mathbb{R}$ functions $f_1, f_2$ (i.e. $exists x : f_1(x) neq f_2(x)$). We have

$E[f_1(x)] = E[f_2(x)]$

Are there conditions that $f_1, f_2$ must satisfy for this to happen?

I translated this problem to integral form as
$ int_{E_1} x dg_1 = int_{E_2} x dg_2$

$g_1, g_2$ being the probability measures of $f_1(x)$ and $f_2(x)$, which can be easily calculated and $E_i$ the corresponding domains. Now, while the domains may be different, they are "similar", so we don't want to just fix domains conveniently -- instead, we want to study $f_1$ and $f_2$. Maybe there's a measure-theoretical backdoor into this, because every lead takes me to functional equations territory, which I can't handle at all.

Disclaimer: Not a homework problem. So yes, I'll be profiting indirectly from the solution, even though it's a tiny piece to a large, mostly non-mathematical puzzle. Also, I hope I'm making myself clear and following the local etiquette. I'm not a native english person, and this is my first post on MO.