Let $\mu$ be a $\sigma$-additive complete measure* defined on the $\sigma$-algebra of the sets of unit $X$. If $g\in L^1(X,\mu)$ is a non-negative function, then, if I am not wrong, we can define a measure $\nu$ having the same characteristics by letting $$\nu (A):=\int_A g\,d\mu$$ for any $\mu$-measurable set $A\subset X$.
I wonder whether the identity $$\int_Xf\,d\nu=\int_X fg\,d\mu$$ holds true for some class of functions $f:X\to\mathbb{R}$ somewhat less trivial than the constant or characteristic functions and, if it does, how can it be proved? I thank any answerer very much.
*Such are the conditions on the measure used to define the Lebesgue integral by A.N. Kolmogorov and S.V. Fomin's Элементы теории функций и функционального анализа, approximately equivalent to Introductory real analysis, which contains all that I know of Lebesgue integration and says nothing directly related to my question.
EDIT (Jul 29 '16): I have been told by a gratuated student in mathematics that $\nu$ may not be complete and Kolmogorov-Fomin's does not define a Lebesgue integral for non-complete measures.
Since it is true for characteristic functions, you can do a simple linearity argument to show it is true for all simple functions. Then do an approximation argument to finish.