mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

— specifically, in

geometric measure theory In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...

— spherical measure ''σ''^''n'' is the "natural"

Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. F ...

on the ''n''-sphere S^''n''. Spherical measure is often normalized so that it is a

probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...

on the sphere, i.e. so that ''σ''^''n''(S^''n'') = 1.

Definition of spherical measure

There are several ways to define spherical measure. One way is to use the usual "round" or " arclength"

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...

''ρ''_''n'' on S^''n''; that is, for points ''x'' and ''y'' in S^''n'', ''ρ''_''n''(''x'', ''y'') is defined to be the (Euclidean) angle that they subtend at the centre of the sphere (the origin of R^''n''+1). Now construct ''n''-dimensional

Hausdorff measure In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that ass ...

''H''^''n'' on the metric space (S^''n'', ''ρ''_''n'') and define :

\sigma^ = \frac H^.

One could also have given S^''n'' the metric that it inherits as a subspace of the Euclidean space R^''n''+1; the same spherical measure results from this choice of metric. Another method uses

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...

''λ''^''n''+1 on the ambient Euclidean space R^''n''+1: for any measurable subset ''A'' of S^''n'', define ''σ''^''n''(''A'') to be the (''n'' + 1)-dimensional volume of the "wedge" in the ball B^''n''+1 that it subtends at the origin. That is, :

\sigma^(A) := \frac \lambda^ ( \ ),

where :

\alpha(m) := \lambda^ (\mathbf_^ (0)).

The fact that all these methods define the same measure on S^''n'' follows from an elegant result of Christensen: all these measures are obviously uniformly distributed on S^''n'', and any two uniformly distributed Borel regular measures on a separable metric space must be constant (positive) multiples of one another. Since all our candidate ''σ''^''n'''s have been normalized to be probability measures, they are all the same measure.

Relationship with other measures

The relationship of spherical measure to Hausdorff measure on the sphere and Lebesgue measure on the ambient space has already been discussed. Spherical measure has a nice relationship to

Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though ...

on the

orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

. Let O(''n'') denote the orthogonal group

acting Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode. Acting involves a broad r ...

on R^''n'' and let ''θ''^''n'' denote its normalized Haar measure (so that ''θ''^''n''(O(''n'')) = 1). The orthogonal group also acts on the sphere S^''n''−1. Then, for any ''x'' ∈ S^''n''−1 and any ''A'' ⊆ S^''n''−1, :

\theta^(\) = \sigma^(A).

In the case that S^''n'' is a

topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...

(that is, when ''n'' is 0, 1 or 3), spherical measure ''σ''^''n'' coincides with (normalized) Haar measure on S^''n''.

Isoperimetric inequality

There is an

isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...

for the sphere with its usual metric and spherical measure (see Ledoux & Talagrand, chapter 1): If ''A'' ⊆ S^''n''−1 is any Borel set and ''B''⊆ S^''n''−1 is a ''ρ''_''n''-ball with the same ''σ''^''n''-measure as ''A'', then, for any ''r'' > 0, :

\sigma^(A_) \geq \sigma^(B_),

where ''A''_''r'' denotes the "inflation" of ''A'' by ''r'', i.e. :

A_ := \.

In particular, if ''σ''^''n''(''A'') ≥ and ''n'' ≥ 2, then :

\sigma^(A_) \geq 1 - \sqrt \, \exp \left( - \frac \right).

References

* * (See chapter 1) * (See chapter 3) {{Measure theory Measures (measure theory)