Convergence in Measure and in Category (Lecture Notes in Mathematics, 2394)

This book exhibits a vast gamut of similarities and differences between measure and (Baire) category.An important similarity is the Sierpinski-Erdos duality theorem: assuming the Continuum Hypothesis, there exists a one-to-one mapping f of the real line R onto itself such that f(A) is a nullset if and only if A is of the first category.Moreover, this mapping can be chosen such that f = f-1. An equally important difference is E. Szpilrajn’s theorem: there does not exist a mapping f of the real line R onto itself such that f(E) is Lebesgue measurable if and only if E has the Baire property.Much of the book is devoted to the study of various modes of convergence: convergence almost everywhere; convergence except on a set of first category; convergence in measure; and convergence in the category of sequences of real functions of a real variable.Here, convergence in a category is *-convergence with respect to convergence except on a set of first category, just as convergence in measure is *