Menshov' "adjustment theorem" with respect to general measures

TL;DR

This paper extends Menshov's theorem, showing that any measurable function can be redefined on a small measure set to have a uniformly convergent Fourier series, even under general Borel measures.

Contribution

The paper generalizes Menshov's adjustment theorem from Lebesgue measure to arbitrary Borel measures, broadening its applicability.

Findings

The theorem holds for any Borel measure, not just Lebesgue measure.

Redefinition on small measure sets ensures uniform convergence of Fourier series.

The result applies to a wide class of measures beyond classical Lebesgue measure.

Abstract

A classical theorem of Menshov states that every measurable function can redefined on a set of arbitrarily small Lebesgue measure, so that the resulting function has uniformly convergent Fourier series. We prove that the same is true if we replace Lebesgue measure with an arbitrary Borel measure.