Borel measurability of separately continuous functions

Lebesgue proved that every separately continuous function f :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 :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, we show that if X is either a countable product of separable linearly ordered spaces, an arbitrary product of countably compact linearly ordered spaces, or the continuous image of an arbitrary product of compact linearly ordered spaces, and Y is any topological space, then every separately continuous function f :X×Y→R is Borel measurable. In the case where X is a product of ordinals, we get stronger results. The results for countably compact linearly ordered spaces use some combinatorial properties of n-dimensional arrays of real numbers which are possibly of independent interest. We also give, under a cardinal arithmetic assumption, an example of a linearly ordered space X and a separately continuous function f :X×X→R which is not Borel measurable.