I need some help, I can't see how to do this, I don't even know how to begin:
If $(X, S,\mu)$ is a measure space and $f$ a nonnegative measurable function on X, let $( f μ)(A) := \int_ A f d\mu$ for any set $A \in S$. If T is measurable and 1–1 from X onto Y for a measurable space (Y,A), with a measurable inverse $T^{-1}$, show that $$( f μ) \circ T^{−1} = ( f \circ T^{-1})(μ \circ T^{−1}).$$
It suffices to show when $f = \chi_E(x)$ a simple function for $E\subset (X,\mathcal{S})$. Let $A\subset(Y,\mathcal{A})$. We see that $$(f\mu)\circ T^{-1} [A] = \int_{T^{-1} [A]} \chi_E(x) d\mu = \int_X \chi_{T^{-1} [A]\cap E}(x) d\mu = \mu (T^{-1} [A]\cap E) \\ = \mu\circ T^{-1}(A\cap T[E]) =\int_Y \textbf{1}_{A\cap T[E]}(y) d(\mu\circ T^{-1}) =\int_Y \textbf{1}_{A} (y)\textbf{1}_{T[E]}(y) \;d(\mu\circ T^{-1}) =\\\int_A {\chi_E} \circ T^{-1} (y) d(\mu\circ T^{-1}) = (f\circ T^{-1})(\mu\circ T^{-1})[A].$$