Measure theoretic description of change-of-variables

1. Settings

Two measurable spaces:

\[(X,\mathcal{A},\mu)\quad\text{and}\quad (Y,\mathcal{B},\mu_T)\]

Consider a transform \(T\) such that:

\[T:X\rightarrow Y\]

where T is a measurable map. i.e., \(T^{-1}(B)\in \mathcal{A}\quad \forall B\in\mathcal{B}\).

\(T\) induces a pushforward measure \(\mu_T\):

\[\mu_T(B)=\mu(T^{-1}(B))\]

2. Change-of-variables

Let \(\nu\) and \(\mu_T\) are measures defined on \(Y\) and \(\nu\) is dominated by \(\mu_T\) (\(\nu\ll\mu_T\)). Then there exists a Radon-Nikodym derivative \(\rho\) such that:

\[\forall B \in \mathcal{B},\quad \nu(B)=\int_B \rho(y)d\mu_T(y) \]

where \(\rho:Y\rightarrow [0,\infty]\). (I missed this point before: The Radon-Nikodym assumes the measures are over the same measurable space.)

Let \(f\) be a nonnegative measurable function on \(Y\). Then,

\[\int_Y f(y)d\mu_T(y)=\int_Xf(T(x))d\mu(x)\]

\[\begin{aligned} \int_Y f(y)d\nu(y)&=\int_Y f(y)\frac{d\nu}{d\mu_T}d\mu_T(y)\\&=\int_Yf(y)\rho(y)d\mu_T(y)\\&=\int_X f(T(x))\rho(T(x))d\mu(x) \end{aligned}\]

3. Examples

• Random variable \(Z: \Omega \rightarrow \mathbb{R}\)
  • A random variable defines a pushforward measure \(\mu_Z\) over the real number line. Assume that the pushforward measure \(\mu_Z\) is dominated by Lebesgue measure\(\lambda\) then \(f=\frac{d\mu_Z}{d\lambda}\). The Radon-Nikodym theorem is applied between the measures \(\mu_T, \lambda\) over the same measurable space \((\mathbb{R}, \mathcal{R})\)
• Inverse transform sampling
• Normalizing flows