Let \((X, \mathscr{A}, \mu)\) be a measure space and \((Y, \mathscr{B})\) be a measurable space. Assume that \(T: A ightarrow B, A \in \mathscr{A}, B \in \mathscr{B}\), is an invertible measurable map. Show that

\[\left.T(\mu)ight|_{B}=T\left(\left.\muight|_{A}ight)\]

with the restrictions \(\left.\muight|_{A}(\cdot):=\mu(A \cap \cdot)\) and \(\left.T(\mu)ight|_{B}:=T(\mu)(B \cap \cdot)\).