The measurable boundaries of an arbitrary function

Abstract

The present paper communicates a number of new properties of arbitrary real functions, of which the most important is the theorem on the measurable boundaries (w 2).This theorem associates with every function f(x) two measurable .functions--its ~)measurable boundaries)) --between which the given function lies almost everywhere, in the sense of measure; moreover, tl~e clustering of the points of y =f(x) is maximal (exterior metric density I) at almost every point of these measurable functions.A general function, in a certain sense, is essentially represented by its measurable boundaries, as is shown by the diverse applications (w 3) of the central theorem of the present paper.This central result, it seems, is the most informative that has appeared concerning the essential internal structure of an arbitrary function.Section ~ gives a useful decomposition of a general set into two components, the one measurable and the other >)homogeneously non-measurable.))This property, though easily derived, seems not to have been explicitly remarked.Section 2 proves.varioustheorems on general functions, including the theorem on the measurable boundaries.Section 3 gives various applications of this theorem, and section 4 briefly discusses several related questions.