Is there a change of variables formula for a measure theoretic integral that does not use the Lebesgue measure

Given a measure space $(X_1,M_1,\mu)$ and a measureable space $(X_2,M_2)$ you can define the pushforward measure on $M_2$ of $\mu$ by a measurable function $F:X_1\to X_2$ to be $F\mu(E)=\mu(F^{-1}(E))$. Then you have the formula

$$\int_{X_2}g\;\mathrm{d}F\mu=\int_{X_1}g\circ F\;\mathrm{d}\mu$$

which is effectively the change of variables between the measure spaces $(X_1,M_1,\mu)$ and $(X_2,M_2,F\mu)$. The change of variables with Lebesgue measure should then a special case of this (the pushforward of $|\mathrm{det} DF|\lambda$ under $F$ is $\lambda$).

Also you can have look on V.I. Bogachev. "Measure Theory."

In the case you are interested in probability theory, see R. Durrett, "Probability: Theory and Examples", 4th ed, 2010, pp 30-31.