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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Banach,  "Sur les lignes rectifiables et les surfaces dont l'aire est finie"  ''Fund. Math.'' , '''7'''  (1925)  pp. 225–236</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Natanson,  "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M.  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Lozinskii,  "On the Banach indicatrix"  ''Vestnik Leningrad. Univ. Math. Mekh. Astr.'' , '''7''' :  2  pp. 70–87  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.S. Kronrod,  "On functions of two variables"  ''Uspekhi Mat. Nauk'' , '''5''' :  1  (1950)  pp. 24–134  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.G. Vitushkin,  "On higher-dimensional variations" , Moscow  (1955)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Banach,  "Sur les lignes rectifiables et les surfaces dont l'aire est finie"  ''Fund. Math.'' , '''7'''  (1925)  pp. 225–236</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Natanson,  "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M.  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Lozinskii,  "On the Banach indicatrix"  ''Vestnik Leningrad. Univ. Math. Mekh. Astr.'' , '''7''' :  2  pp. 70–87  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.S. Kronrod,  "On functions of two variables"  ''Uspekhi Mat. Nauk'' , '''5''' :  1  (1950)  pp. 24–134  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.G. Vitushkin,  "On higher-dimensional variations" , Moscow  (1955)  (In Russian)</TD></TR></table>
 
  
  
 
====Comments====
 
====Comments====
More generally, for any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515018.png" /> define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515019.png" /> analogously. Then, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515020.png" /> be a separable metric space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515021.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515022.png" />-measurable for all Borel subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515024.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515025.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515026.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515027.png" /> be the measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515028.png" /> defined by the Carathéodory construction from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515029.png" />. Then
+
More generally, for any mapping $f:X\to Y$ define $N(y,f)$ analogously.  
 
+
Then, let $X$ be a separable metric space and let $f(A)$ be $\mu$-measurable for all Borel subsets $A$ of $X$.  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515030.png" /></td> </tr></table>
+
Let $\zeta(S) = \mu(f(S))$ for $S\subset X$ and let $\psi$ be the measure on $X$ defined by the Carathéodory construction from $\zeta$. Then
 
+
$$
for every Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015150/b01515031.png" />. Cf. [[#References|[a1]]], p. 176 ff. For significant extension of (*), cf. [[#References|[a2]]].
+
\psi(A) = \int\limits_{A}N(y,f)\, d\mu_{Y}
 +
$$
 +
for every Borel set $A\subset X$. Cf. [[#References|[a1]]], p. 176 ff. For significant extension of \eqref{eq1}, cf. [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Federer,  "Geometric measure theory" , Springer  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Federer,  "An analytic characterization of distributions whose partial derivatives are representable by measures"  ''Bull. Amer. Math. Soc.'' , '''60'''  (1954)  pp. 339</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Federer,  "Geometric measure theory" , Springer  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Federer,  "An analytic characterization of distributions whose partial derivatives are representable by measures"  ''Bull. Amer. Math. Soc.'' , '''60'''  (1954)  pp. 339</TD></TR></table>

Latest revision as of 14:32, 16 May 2015

multiplicity function, of a continuous function $y=f(x)$, $a\leq x\leq b$

An integer-valued function $N(y,f)$, $-\infty < y < \infty$, equal to the number of roots of the equation $f(x)=y$. If, for a given value of $y$, this equation has an infinite number of roots, then $$ N(y,f) = +\infty, $$

and if it has no roots, then

$$ N(y,f) = 0. $$

The function $N(y,f)$ was defined by S. Banach [1] (see also [2]). He proved that the indicatrix $N(y,f)$ of any continuous function $f(x)$ in the interval $[a,b]$ is a function of Baire class not higher than 2, and \begin{equation}\label{eq1} V_a^b(f) = \int\limits_{-\infty}^{+\infty} N(y, f) \, dy, \end{equation}

where $V_a^b(f)$ is the variation of $f(x)$ on $[a,b]$. Thus, equation \eqref{eq1} can be considered as the definition of the variation of a continuous function $f(x)$. The Banach indicatrix is also defined (preserving equation \eqref{eq1}) for functions with discontinuities of the first kind [3]. The concept of a Banach indicatrix was employed to define the variation of functions in several variables [4], [5].

References

[1] S. Banach, "Sur les lignes rectifiables et les surfaces dont l'aire est finie" Fund. Math. , 7 (1925) pp. 225–236
[2] I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)
[3] S.M. Lozinskii, "On the Banach indicatrix" Vestnik Leningrad. Univ. Math. Mekh. Astr. , 7 : 2 pp. 70–87 (In Russian)
[4] A.S. Kronrod, "On functions of two variables" Uspekhi Mat. Nauk , 5 : 1 (1950) pp. 24–134 (In Russian)
[5] A.G. Vitushkin, "On higher-dimensional variations" , Moscow (1955) (In Russian)


Comments

More generally, for any mapping $f:X\to Y$ define $N(y,f)$ analogously. Then, let $X$ be a separable metric space and let $f(A)$ be $\mu$-measurable for all Borel subsets $A$ of $X$. Let $\zeta(S) = \mu(f(S))$ for $S\subset X$ and let $\psi$ be the measure on $X$ defined by the Carathéodory construction from $\zeta$. Then $$ \psi(A) = \int\limits_{A}N(y,f)\, d\mu_{Y} $$ for every Borel set $A\subset X$. Cf. [a1], p. 176 ff. For significant extension of \eqref{eq1}, cf. [a2].

References

[a1] H. Federer, "Geometric measure theory" , Springer (1969)
[a2] H. Federer, "An analytic characterization of distributions whose partial derivatives are representable by measures" Bull. Amer. Math. Soc. , 60 (1954) pp. 339
How to Cite This Entry:
Banach indicatrix. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Banach_indicatrix&oldid=36412
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article