On measurability of Banach indicatrix

TL;DR

This paper proves the measurability of the Banach indicatrix, a multiplicity function, for measurable mappings between metric measure spaces, addressing a question previously discussed online.

Contribution

It establishes the measurability of the Banach indicatrix for measurable functions between metric measure spaces, filling a gap in the mathematical literature.

Findings

Proves the measurability of the Banach indicatrix.

Addresses an open question from MathOverflow.

Provides a rigorous foundation for multiplicity functions in metric measure spaces.

Abstract

Given two metric measure spaces $X$ and $Y$. Let $f:X\to Y$ be a measurable mapping and $A\subset X$. The Banach indicatrix (multiplicity function) is defined as $N(y,f,A) = \#\{x\in A \mid f(x) = y\}$. We prove measurability of this multiplicity function. The question was also discussed on http://mathoverflow.net/q/206780/15946.