(Change of variables formula). Let ( $X, \mathcal{B}, \mu$ ) be a measure space, and let $\phi: X \rightarrow Y$ be a measurable morphism (as defined in Remark 1.4.33) from ( $X, \mathcal{B}$ ) to another measurable space $(Y, \mathcal{C})$. Define the pushforward $\phi_* \mu: \mathcal{C} \rightarrow[0,+\infty]$ of $\mu$ by $\phi$ by the formula $\phi_* \mu(E):=\mu\left(\phi^{-1}(E)\right)$.
(i) Show that $\phi_* \mu$ is a measure on $\mathcal{C}$, so that $\left(Y, \mathcal{C}, \phi_* \mu\right)$ is a measure space.
(ii) If $f: Y \rightarrow[0,+\infty]$ is measurable, show that $\int_Y f d \phi_* \mu= \int_X(f \circ \phi) d \mu$.
(Hint: the quickest proof here is via the monotone convergence theorem (Theorem 1.4.44) below, but it is also possible to prove the exercise without this theorem.)