Sums, products and continuity of Borel maps in nonseparable metric spaces

Abstract:

We give a comparatively simple proof that the complex map $(f,g):T \to X \times Y$ will be (Borel measurable) of class $\alpha + \alpha$, whenever $f$ and $g$ are of class $\alpha$, for not necessarily separable metric spaces $T,X$ and $Y$. The Borel measurability of other types of maps, such as the sum of two vector-valued maps, is easily deduced from this. A general version of this result, applicable to abstract measurable spaces, is also proven. Our second principal result shows that, if $f:T \times X \to Y$ is continuous in each variable separately, where $X$ and $Y$ are metrizable, then $f$ has a $\sigma$-locally finite function base of closed sets, and thus will be continuous at each point of the complement of some first category subset of $T \times X$.