Spherical measure

In mathematics — specifically, in geometric measure theory — spherical measure σn is the “natural” Borel measure on the n-sphere Sn. Spherical measure is often normalized so that it is a probability measure on the sphere, i.e. so that σn(Sn) = 1.There are several ways to define spherical measure. One way is to use the usual “round” or “arclength” metric ρn on Sn; that is, for points x and y in Sn, ρn(x, y) is defined to be the (Euclidean) angle that they subtend at the centre of the sphere (the origin of Rn+1). Now construct n-dimensional Hausdorff measure Hn on the metric space (Sn, ρn) and defineThe relationship of spherical measure to Hausdorff measure on the sphere and Lebesgue measure on the ambient space has already been discussed.There is an isoperimetric inequality for the sphere with its usual metric and spherical measure (see Ledoux & Talagrand, chapter 1):