Is there a change of variables formula for a measure theoretic integral that does not use the Lebesgue measure Is there a generic change of variables formula for a measure theoretic integral that does not use the Lebesgue measure?  Specifically, most references that I can find give a change of variables formula of the form:
$$
\int_{\phi(\Omega)} f d\lambda^m = \int_{\Omega} f \circ \phi |\det J_\phi| d\lambda^m
$$
where $\Omega\subset\Re^m$, $\lambda^m$ denotes the $m$-dimensional Lebesgue measure, and $J_\phi$ denotes the Jacobian of $\phi$.  Is it possible to replace $\lambda^m$ with a generic measure and, if so, is there a good reference for the proof?  I'm also curious if a similar formula holds in infinite dimensions.
 A: If $(X_1,F_1)$ and $(X_2,F_2)$ are two measurable space,
$$T:X_1\to X_2$$
is a measurable function, and
$$\mu:F_1\to[0,\infty]$$
is a measure on $X_1$, then the mesaure
$$\mu \circ T^{-1}:X_2\to [0,\infty]$$
defined by
$$\mu\circ T^{-1}(B)=\mu(T^{-1}(B))\qquad B\in F_2$$
is a measure on $X_2$. A measurable function $f$ on $X_2$ is integrable with respect to $\mu\circ T^{-1}$ iff $f\circ T$ is integrable with respect to $\mu$ and we have

$$\int_{X_2}f\;d(\mu\circ T^{-1})=\int_{X_1}f\circ T\;d\mu. \tag{1}$$

If $T$ is one-to-one, we have for any $A\in F_1$:

$$\int_{T(A)}f\;d(\mu\circ T^{-1})=\int_{A}f\circ T\;d\mu \tag{2}$$

Proof of (1)
It is clear that $\mu\circ T^{-1}$ is measure on $X_2$.
For proof of (1) it suffices to show it holds for any nonnegative real valued function $f$ and so it is proved for every real measurable function since:
$$f=f^+-f^-.$$
Equation (1) is proved for any complex measurable function $(f=f_{Re}+i\,f_{Im})$.
Let $f$ be a nonnegative real-valued measurable function.
Then there exists a sequence of simple measurable functions $\phi_n=\sum_{i=1}^{m_n} c_{n,i}\,\chi_{B_{n,i}}$ such that
$$\phi_n\nearrow f$$
and since the integral of left side of (1) equals to:
$$
\begin{align*}
&\int_{X_2}f\;\;d(\mu\circ T^{-1}) \\
&=\lim_{n\to \infty}\int_{X_2}\phi_n \;d(\mu\circ T^{-1}) \\
&=\lim_{n\to \infty}\int_{X_2}\sum_{i=1}^{m_n}c_{n,i}\,\chi_{B_{n,i}} \;d(\mu\circ T^{-1}) \\
&=\lim_{n\to\infty}\sum_{i=1}^{m_n}c_{n,i}\, \int_{X_2}\chi_{B_{n,i}} \;d(\mu\circ T^{-1})\\
&=\lim_{n\to\infty}\sum_{i=1}^{m_n}c_{n,i}\;\mu(T^{-1}(B_{n,i}))\\
&=\lim_{n\to\infty}\sum_{i=1}^{m_n}c_{n,i}\;\int_{X_1}\chi_{T^{-1}(B_{n,i})}\;d\mu\\
&=\lim_{n\to\infty}\sum_{i=1}^{m_n}c_{n,i}\;\int_{X_1}\chi_{B_{n,i}}\circ T\;\;d\mu \\
&=\lim_{n\to \infty}\int_{X_1}\left[\sum_{i=1}^{m_n}c_{n,i}\,\chi_{B_{n,i}}\right]\circ T\;\; d\mu \\ &=\lim_{n\to \infty}\int_{X_1}\phi_n\circ T\;\; d\mu \\
&=\int_{X_1}f\circ T\;\; d\mu.
\end{align*}
$$
The last paragraph is established because $\phi_n\circ T$ is a simple measurable function such that
$$\phi_n\circ T\;\nearrow\; f\circ T.$$
Proof of (2)
Now for proof of (2), we have:
$$
\begin{align*}
\int_{T(A)}f\;\;d(\mu\circ T^{-1})
&=\int_{X_2}f\;\chi_{T(A)}\;\;d(\mu\circ T^{-1}) \\
&=\int_{X_1}(f\;\chi_{T(A)})\circ T\quad d\mu \\
&\color{magenta}{=}\int_{X_1}(f\circ T)\,.\,\chi_{A}\quad d\mu \\
&=\int_{A}f\circ T\quad d\mu.
\end{align*}
$$
The equality in pink holds because
$$
\begin{align*}
&(f\;\chi_{T(A)})\circ T\,(x) \\
&= f(T(x))\;.\chi_{T(A)}(T(x)) \\
&= \begin{cases}
  f(T(x))  & \text{ $T(x)\in T(A)$} \\
  0  & \text{ $T(x)\notin T(A)$}
\end{cases} \\
&\color{red}{=}\begin{cases}
  f(T(x))    &      \text{ $x\in A$} \\
  0          &      \text{ $x\notin A$}
\end{cases} \\
&=[\,(f\circ T)\,.\,\chi_A\,](x)
\end{align*}
$$
where the equality in red holds because $T$ is one to one.
A: 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.
A: 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 $\lambda$).
A: please have a look at the monograph by Patric Muldowney theory of Random variation John Wiley and sons. it suggests a formuala and proves using Henstock-kurzweil apparoach
