Let $M_X = (X, \mathcal{F}_X)$ and $M_Y = (Y, \mathcal{F}_Y)$ each be a D1108: Measurable space.
A D18: Map $f : X \to Y$ is a measurable space isomorphism from $M_X$ to $M_Y$ if and only if

(1)	$f$ is a D201: Measurable map from $M_X$ to $M_Y$
(2)	$f$ is an D976: Invertible map with an D216: Inverse map $f^{-1} : Y \to X$
(3)	$f^{-1}$ is a D201: Measurable map from $M_Y$ to $M_X$