Borel measurability of separately continuous functions
by
Maxim R. Burke
University of Prince Edward Island

Lebesgue proved that every separately continuous function f\colon R×R --> R is a pointwise limit of continuous functions. W. Rudin extended this by showing that if X is a metric space, then for any topological space Y, every separately continuous function f\colon X×Y --> R is a pointwise limit of continuous functions. This statement can fail if we take for X an arbitrary linearly ordered space, even if X is separable. However, if X is either a product of at most continuum many separable linearly ordered spaces, or an arbitrary product of countably compact linearly ordered spaces, and Y is any topological space, then every separately continuous function f\colon X×Y --> R is Borel measurable. If the countably compact spaces in the last statement are compact, then the conclusion follows by a simpler argument from the following theorem which extends the same result for a single factor due to Haydon, Jayne, Namioka and Rogers: when K is a product of compact linearly ordered spaces, C(K) has a pointwise Kadec renorming, i.e., there is a norm equivalent to the sup norm for which the norm topology and the topology of pointwise convergence coincide on the unit sphere. Some of these results are part of joint work with W. Kubis and S. Todorcevic.

Date received: June 3, 2003