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$f : [a,b] \to \mathbb R$ 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?

user277793
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  • This an exercise from W. Rudin book "Real and Complex analysis" (exercise 8.22) and there in this book is an hint how to prove it. –  Nov 27 '15 at 10:30
  • I can't find exercise 8.22 in the book. Isn't it 2nd edition? If you have the book now, can you let me know the hint? Please – user277793 Nov 29 '15 at 07:18
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    The hint is; "The theorem is obvious for functions whose graphs are union of line segments. But every continuous function can by approximated by such functions" –  Nov 29 '15 at 13:35
  • Thank you . Now I can prove it with your hint – user277793 Dec 02 '15 at 08:57
  • If you use simple functions (step functions), $M(y)$ will be $+\infty$ at finitely many points, and zero elsewhere. I don't think this will work. For example, if $f(x)=x$, then $M(y)=1$, there is no way you will reach this with a function taking only the values $+\infty$ and $0$. I believe Rudin means you should use functions which are locally linear (resp. locally affine) – Idontgetit Mar 18 '21 at 20:22
  • Here is some idea (just an idea). Fix a $y$ and for now assume that it is finite. place tiny open sets around these points. Show that for z near y these balls also cover preimage of z. By taking limit you end up proving semi-continuity of the function. – Behnam Esmayli Mar 22 '21 at 00:04

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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

Yuval Peres
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