INFINITE COMBINATORICS IN FUNCTION SPACES: CATEGORY METHODS

N. H. Bingham and A. J. Ostaszewski

Abstract: The infinite combinatorics here give statements in which, from some sequence, an infinite subsequence will satisfy some condition – for example, belong to some specified set. Our results give such statements generically – that is, for `nearly all' points, or as we shall say, for quasi all points – all off a null set in the measure case, or all off 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, which contains the Kestelman–Borwein–Ditor Theorem 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.

Keywords: automatic continuity, measurable function, Baire property, generic property, infinite combinatorics, function spaces, additive function, subadditive function, mid-point convex function, regularly varying function

Classification (MSC2000): 26A03

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