APPROXIMATE MONOTONICITY - THEORY AND APPLICATIONS

The ideas of value distribution for measurable functions from pg to R are applied to functions which are approximately monotonic on sets of positive measure. (For definitions see 1.) A function p(x) is introduc ed, describing the local relative value distribution in the neighbourh ood of a point x, and it is shown that almost everywhere p(x) = 0 or 1 /2 wherever p(x) exists, implying approximate differentiabiilty, with the function approximately oscillatory elsewhere. These results are ap plied to the analysis of angular boundary behaviour for Herglotz funct ions, where they have implications for the spectral analysis of differ ential and other operators.