Let \((X, \mathscr{A}, \mu)\) be a measure space and let \(T: X ightarrow X\) be a bijective measurable map whose inverse \(T^{-1}: X ightarrow X\) is again measurable. Show that for every \(f \in \mathcal{M}^{+}(\mathscr{A})\) one has

\[\int u d(T(f \mu))=\int u \circ T f d \mu=\int u f \circ T^{-1} d T(\mu)=\int u d\left(f \circ T^{-1} T(\mu)ight)\]