View source for Hausdorff measure

A collective name for the class of measures determined on the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467001.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467002.png" /> of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467003.png" /> by means of the following construction: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467004.png" /> be a certain class of open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467005.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467006.png" /> be a non-negative function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h0467007.png" />, and let

<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/h/h046/h046700/h0467008.png" /></td> </tr></table>

<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/h/h046/h046700/h0467009.png" /></td> </tr></table>

where the infimum is taken over all finite or countable coverings of the Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670010.png" /> by sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670011.png" /> with diameter not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670012.png" />. The Hausdorff measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670013.png" /> defined by the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670014.png" /> and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670015.png" /> is the limit

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Examples of Hausdorff measures. 1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670017.png" /> be the collection of all balls in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670018.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670020.png" />. The corresponding measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670021.png" /> is called the Hausdorff <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670023.png" />-measure (the linear Hausdorff measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670024.png" /> and the plane Hausdorff measure for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670025.png" />). 2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670026.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670027.png" /> be the collection of cylinders with spherical bases and with axes parallel to the direction of the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670028.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670029.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670030.png" />-dimensional volume of the axial section of a cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670031.png" />; the corresponding Hausdorff measure is called the cylindrical measure.

The Hausdorff measures were introduced by F. Hausdorff [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Hausdorff,   "Dimension and äusseres Mass"  ''Math. Ann.'' , '''79'''  (1918)  pp. 157–179</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR></table>



====Comments====
A method to construct measures on metric spaces was introduced by C. Carathéodory in 1914. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670032.png" /> can be anything and are often taken closed. The Hausdorff measures are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670033.png" />-additive on the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670034.png" />-field, but are in general not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670035.png" />-finite; some restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670038.png" /> have to be added in order to get nice properties of approximation from below. Such restrictions are, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670039.png" /> is a Borel subset of some compact metric space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670040.png" /> is the class of closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670042.png" /> is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670043.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670044.png" /> is a continuous non-decreasing function from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670045.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670046.png" />. The measures obtained in this way are the Hausdorff measures most often used, and were mainly studied by A.S. Besicovitch and his school (cf. [[#References|[a8]]]); they are called (Hausdorff) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670047.png" />-measures (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670049.png" />, one says <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670050.png" />-measure or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670051.png" />-dimensional measure; see also [[Hausdorff dimension|Hausdorff dimension]]). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670052.png" /> is the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670053.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670054.png" />-dimensional measure is equal (up to a multiplicative constant) to the [[Lebesgue measure|Lebesgue measure]] if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670055.png" /> and, when restricted to smooth curves, surfaces, etc., is equal to the length, area, etc. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670056.png" />, etc. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670057.png" />-measure is the counting measure, which is also at the confines of potential theory and descriptive set theory.

Even though the notion of Hausdorff measure is not regarded as being fundamental, it appears in many parts of hard analysis and geometry. Through the study of exceptional sets, it is used, for example, in [[Harmonic analysis|harmonic analysis]] (cf. [[#References|[a5]]]), in [[Potential theory|potential theory]] (cf. [[#References|[a1]]]), in the metric theory of continued fractions (cf. [[#References|[a8]]]), and in differential geometry (cf. [[Sard theorem|Sard theorem]]). For many reasons it is closely linked to the notion of [[Capacity|capacity]], and, more generally, to [[Descriptive set theory|descriptive set theory]] (cf. [[#References|[a1]]], [[#References|[a2]]], [[#References|[a8]]]; R.O. Davies was, before G. Choquet, the first to prove capacitability theorems). In the theory of stochastic processes it has a crucial role in the fine study of the paths of the [[Wiener process|Wiener process]] and others (cf. [[#References|[a6]]] and its bibliography). Finally, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670058.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670059.png" />-dimensional measure is, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670060.png" /> is a natural integer, a fundamental notion in geometric measure theory (cf. [[Area|Area]]; [[Minimal surface|Minimal surface]], and [[#References|[a4]]] in which also Hausdorff measures which are not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670061.png" />-measures ([[Favard measure|Favard measure]], etc.) are used); for non-integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046700/h04670062.png" /> it is a fundamental notion in the theory of [[Fractals|fractals]] (cf. [[#References|[a3]]]).

For the use of Hausdorff measures and the [[Hausdorff dimension|Hausdorff dimension]] in multi-dimensional complex analysis, see, e.g., [[#References|[a9]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Carleson,   "Selected problems on exceptional sets" , v. Nostrand  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Dellacherie,   "Ensembles analytiques, capacités, mesures de Hausdorff" , ''Lect. notes in math.'' , '''295''' , Springer  (1972)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K.J. Falconer,   "The geometry of fractal sets" , Cambridge Univ. Press  (1985)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Federer,   "Geometric measure theory" , Springer  (1969)  pp. 60; 62; 71; 108</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.-P. Kahane,   R. Salem,   "Ensembles parfaits et séries trigonométriques" , Hermann  (1963)  pp. 142</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.-F. le Gall,   "Temps locaux d'intersection et points multiples des processus de Lévy" , ''Sem. Probab. XXI'' , ''Lect. notes in math.'' , '''1247''' , Springer  (1987)  pp. 341–374</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M.E. Munroe,   "Introduction to measure and integration" , Addison-Wesley  (1953)  pp. 111</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  C.A. Rogers,   "Hausdorff measures" , Cambridge Univ. Press  (1970)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E.M. Chirka,   "Complex analytic sets" , Kluwer  (1989)  (Translated from Russian)</TD></TR></table>