Problem with almost everywhere equality

A topological space $Y$ is said to have (AEEP) if the following condition is fulfilled. Whenever $(X,\mathfrak{M})$ is a measurable space and $f, g: X \to Y$aretwomeasurablefunctions,thentheset$\Delta(f,g) = {x \in X:\ f(x) = g(x)}$isamemberof$\mathfrak{M}$\mathrm{.}Itisshownthatametrizablespace$Y$has(AEEP)iffthecardinalityof$Y$isnogreaterthan$2^{\aleph_0}$.