Characterizing isotropic measures A Borel measure $\mu$ on $S^{n-1}$ is called isotropic if $$\int_{S^{n-1}}
\langle \theta, x \rangle^2 d\mu(x)=\frac{\mu\left({S^{n-1}}\right)}{n}$$ for all $\theta\in S^{n-1}$.
This means that in some sense the measure is "evenly distributed" on the sphere, but I'm wondering if one can give more explicit characterizations of such measures, or some geometric intuition, possibly in terms of the support. For instance, an isotropic measure cannot be supported on a subset of a proper subspace, because then there is some $\theta$ for which the integral vanishes. Are there other such simple necessary conditions?
Some examples are also useful: for instance, the Lebesgue measure is isotropic. On $S^1$ it can be shown with some trigonometry that a discrete uniform measure on the points $\textrm{exp}\left(\frac{\pi i j}{k}\right)$ is isotropic. These examples give an idea of the "even distribution" of these measures (and their support), but I'm really hoping more can be said.  For instance, is there a similar set of discrete isotropic measures in higher dimensions? How can we show this? How about other (discrete) uniform measures? etc. 
 A: One way to obtain such isotropic measures, at least on $S^2$ and probably on higher-dimensional spheres up to some topological identifications, is the following, where I always take $n=3$.
On the $2$-torus there are plenty measures with the "uniform" distribution that you describe: for example take any translation with irrational slope. Then you can project this to the sphere by taking a branched cover (with four branch points). As far as I understand the induced measure will have the property that you want.
A: If you take an orthonormal basis $E=\{e_1,\ldots,e_n\}$ on the sphere $\mathbb{S}^{n-1}$, the measure 
$$\mu=\sum_{i=1}^n \delta_{e_i}$$
is isotropic, because the integral gives $|\theta|^2$, and so is any finite linear combination of such measures obtained by different o.n. bases.
You can also consider an integral average of $\mu$. If $\Theta$ is a probability on the space of rotations $O(n)$, you can define
$$\mu_\Theta =\int\limits_{O(n)}(T_\sharp \mu)d\Theta(T)$$
where $T_\sharp \mu$ is the pushforward measure given by $T_\sharp \mu(A)=\mu(T^{-1}(A))$. Then
\begin{align}
\int\limits_{\mathcal{S}^{n-1}}\langle \theta,x\rangle^2d\mu_\Theta(x)&= \int_{O(n)}d\Theta(T)\int\limits_{\mathcal{S}^{n-1}}\langle \theta,x\rangle^2dT_\sharp\mu(x)= \\
&=\int_{O(n)}d\Theta(T)\int\limits_{\mathcal{S}^{n-1}}\langle \theta,T^{-1}x\rangle^2d\mu(x)\\
&=\int_{O(n)}d\Theta(T)\int\limits_{\mathcal{S}^{n-1}}\langle T\theta,x\rangle^2d\mu(x)\\
&=\frac{\mu(\mathcal{S}^{n-1})}{n}=\frac{\mu_\Theta(\mathcal{S}^{n-1})}{n}
\end{align}
so $\mu_\Theta$ is isotropic, and this works starting from any isotropic $\mu$.
