# Spherical measure

In mathematics — specifically, in geometric measure theory — spherical measure σ$^n$ is the "natural" Borel measure on the n-sphere S$^n$. Spherical measure is often normalized so that it is a probability measure 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 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 H$^n$ on the metric space (S$^n$, ρ$_n$) and define
    $\sigma ^{n}={\frac {1}{H^{n}(\mathbf {S} ^{n})}}H^{n}.$
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 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,
    $\sigma ^{n}(A):={\frac {1}{\alpha (n+1)}}\lambda ^{n+1}(\{tx\mid x\in A,t\in [0,1]\}),$
where
    $\alpha (m):=\lambda ^{m}(\mathbf {B} _{1}^{m}(0)){\text{ }}(\mathbf {B} _{1}^{m}(0){\text{ is the ball of radius 1 centered at the origin in }}$R$^{n+1}$).
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 on the orthogonal group. Let O(n) denote the orthogonal group acting 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$^{n−1}$,
    $\theta ^{n}(\{g\in \mathrm {O} (n)\mid g(x)\in A\})=\sigma ^{n-1}(A).$
In the case that S$^n$ is a topological group (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 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,
    $\sigma ^{n}(A_{r})\geq \sigma ^{n}(B_{r}),$
where A$_r$ denotes the "inflation" of A by r, i.e.
    $A_{r}:=\{x\in \mathbf {S} ^{n}\mid \rho _{n}(x,A)\leq r\}.$
In particular, if σ$^n$(A) ≥ ⁠1/2⁠ and n ≥ 2, then
    $\sigma ^{n}(A_{r})\geq 1-{\sqrt {\frac {\pi }{8}}}\,\exp \left(-{\frac {(n-1)r^{2}}{2}}\right).$