I have the relationship $\int_E h d\mu = \int_E d\nu$ for some integrable function $h$ and measures $\nu$ and $\mu$. I then multiply both sides of the equation with some other integrable function $g$ and then plug it inside the integrals to get
$$\int_Eg h d\mu = \int_E gd\nu$$
It seems intuitive to me but is that allowed for any $g$? I know I can do it with constants and Lebesgue integrals, but I'm not sure it's allowed for functions. Any ideas?
No, that is not possible. For instance $$ \int_0^{2\pi}\sin x\,dx=\int_0^{2\pi}0\,dx=0, $$ but $$ \int_0^{2\pi}\sin x\times\sin x\,dx\ne\int_0^{2\pi}\sin x\times0\,dx. $$