Image Measure

In the setting of Measurable Functions and the Lebesgue Integral: Notation, with its measure space $(X,\mathcal{F},\mu)$, let $(Y,\mathcal{G})$ be a measurable space, and let $T\in\mathrm{Meas}(X,\mathcal{F};Y,\mathcal{G})$.

(i) Image measure. The image measure of $\mu$ under $T$ is the function $\mu\circ T^{-1}:\mathcal{G}\to[0,\infty]$ with

$$(\mu\circ T^{-1})(B)=\mu\bigl(T^{-1}(B)\bigr)\qquad(B\in\mathcal{G}).$$

(Here $T^{-1}(B)\in\mathcal{F}$ because $T$ is $\mathcal{F}/\mathcal{G}$-measurable, and $\mu\circ T^{-1}$ is the composite of $\mu$ with the restriction to $\mathcal{G}$ of the preimage function of Preimages under a Map: Membership, the Preimage Map, Sequences of Preimages, and the Set of Sets with Preimage in a Given Set (iii).)