On $\sigma$-algebras related to the measurability of compositions

I. V. Shragin

Matematičeskie zametki, Tome 80 (2006) no. 6, pp. 926-933

Résumé

Given a measurable space $(T,\mathscr T)$, a set $X$, and a map $\varphi\colon T\to X$, the $\sigma$-algebras $$ \mathfrak N_\varphi=\{B\subset X:\varphi^{-1}(B)\in\mathscr T\},\qquad \mathfrak M_\varphi=\{D\subset T\times X:G_\varphi^{-1}(D)\in\mathscr T\}, $$ $\mathfrak N_\Phi=\bigcap_{\varphi\in\Phi}\mathfrak N_\varphi$, and $\mathfrak M_\Phi=\bigcap_{\varphi\in\Phi}\mathfrak M_\varphi$, where $G_\varphi(t)=(t,\varphi(t))$ and $\Phi\subset X^T$, are considered. These $\sigma$-algebras are used to characterize the $(\mathscr T,\mathscr B)$-measurability of the compositions $g\circ\varphi$ and $f\circ G_\varphi$, where $g\colon X\to Y$, $f\colon T\times X\to Y$, and $(Y,\mathscr B)$ is a measurable space. Their elements are described without using the operations $\varphi^{-1}$ and $G_\varphi^{-1}$.