Infinite Combinatorics in Function Spaces: Category Methods

The in…nite combinatorics here give statements in which, from some sequence, an in…nite subsequence will satisfy some condition – for example, belong to some speci…ed set. Our results give such statements generically –that is, for ‘nearly all’points, or as we shall say, for quasi all points –all o¤ a null set in the measure case, or all o¤ a meagre set in the category case. The prototypical result here goes back to Kestelman in 1947 and to Borwein and Ditor in the measure case, and can be extended to the category case also. Our main result is what we call the Category Embedding Theorem (CET), which contains the Kestelman-Borwein-Ditor Theorem (KBD) as a special case. Our main contribution is to obtain functionwise rather than pointwise versions of such results. We thus subsume results in a number of recent and related areas, concerning e.g. additive, subadditive, convex and regularly varying functions. Classi…cation: 26A03 Keywords: automatic continuity, measurable function, Baire property, generic property, in…nite combinatorics, function spaces, additive function, subadditive function, mid-point convex function, regularly varying function.