Integrable Function under 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 mapping.

Then the following are equivalent:


 * $(1): \quad f$ is $\map T \mu$-integrable
 * $(2): \quad f \circ T$ is $\mu$-integrable

where $\map T \mu$ is the pushforward measure of $\mu$ under $T$.