Represent total variation of continuous function by integration of counting function

Question

$f : [a,b] \to \mathbb R$ is continuous, let $M(y)$ be the number of points $x$ in $[a,b]$ such that $f(x)=y$. prove that $M$ is Borel masurable and $\int M(y)dy$ equals the total variation of $f$ on $[a,b]$

This is an exercise in 'real analysis for graduate students'(Richard.F.Bass)

At first time, I thought $M(y)=\mu(\{x \mid f(x)=y\})$ where $\mu$ is counting measure. But I cannot find the relation between Borel measurability and $\mu(\{x \mid f(x)=y\})$

Is there anyone would help me?

Answer 1

This is the Banach indicatrix Theorem, which he proved in 1925:

[1] S. Banach, "Sur les lignes rectifiables et les surfaces dont l'aire est finie" Fund. Math. , 7 (1925) pp. 225–236 http://matwbn.icm.edu.pl/ksiazki/fm/fm7/fm7116.pdf

An exposition in English of Banach's proof is given in

Banach Indicatrix Function

Generalizations are in

[2] S.M. Lozinskii, "On the Banach indicatrix" Vestnik Leningrad. Univ. Math. Mekh. Astr. , 7 : 2 pp. 70–87 (In Russian)

[3] https://projecteuclid.org/journals/real-analysis-exchange/volume-27/issue-2/Generalization-of-the-Banach-Indicatrix-Theorem/rae/1212412867.full