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 define One could also have given Sn the metric that it inherits as a subspace of the Euclidean space Rn+1; the same spherical measure results from this choice of metric.
Another method uses Lebesgue measure λn+1 on the ambient Euclidean space Rn+1: for any measurable subset A of Sn, define σn(A) to be the (n + 1)-dimensional volume of the "wedge" in the ball Bn+1 that it subtends at the origin.
That is, where The fact that all these methods define the same measure on Sn follows from an elegant result of Christensen: all these measures are obviously uniformly distributed on Sn, and any two uniformly distributed Borel regular measures on a separable metric space must be constant (positive) multiples of one another.
There is an isoperimetric inequality for the sphere with its usual metric and spherical measure (see Ledoux & Talagrand, chapter 1): If A ⊆ Sn−1 is any Borel set and B⊆ Sn−1 is a ρn-ball with the same σn-measure as A, then, for any r > 0, where Ar denotes the "inflation" of A by r, i.e.