A New Example of First Class Function
by
Ondřej Kalenda

Let X be a topological space, M a metric space. The map f : X --> M is called of the first Borel class if it is (F /\ G)_\sigma-measurable, i.e. f^-1(U) is a countable union of sets of the form F \cap G with G open and F closed in X for every U open subset of M. The map f is said to have the point of continuity property if f\restriction F has a continuity point for every F subset X nonempty closed.

A classical result of Baire states that if X and M are Polish spaces, then every function of the first Borel class of X to M has the point of continuity property.

Koumoullis (1993) proved that if the space of Radon probability measures on every nonempty closed subspace of X is a Baire space when endowed with the weak topology (X is hereditarily t-Baire) and if there is no measurable cardinal, then every function of the first Borel class defined on X has the point of continuity property, and he asked if the set-theoretic assumption is necessary.

The answer is yes because the following holds:

Theorem If there is a measurable cardinal, then there is a hereditarily t-Baire space X and an (F /\ G)-measurable function on X which has no point of continuity.

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-49.