# Integral with respect to Pushforward Measure

## Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\struct {X', \Sigma'}$ be a measurable space.

Let $T: X \to X'$ be a $\Sigma \, / \, \Sigma'$-measurable mapping.

Let $f: X' \to \overline \R$ be a $\map T \mu$-integrable function, where $\map T \mu$ denotes the pushforward measure of $\mu$ under $T$.

Then $f \circ T: X \to \overline \R$ is $\mu$-integrable, and:

$\displaystyle \int_{X'} f \rd \map T \mu = \int_X f \circ T \rd \mu$