= Radial disintegration of an orthogonally invariant measure
{title2=$\int f\,d\mu=\int_{[0,\infty)}\int_{\mathbb S^{d-1}}f(r\theta)\,d\sigma(\theta)\,d(|\cdot|_\#\mu)(r)$}
An <orthogonally invariant random vector> has a <probability measure> determined by its radial <pushforward measure>. Here $\sigma$ is normalized <surface measure on a sphere>, and the formula holds for bounded measurable $f$ and for nonnegative measurable $f$. It includes an <atom of a measure> at zero: the inner integral there is simply $f(0)$.
To prove the formula first for continuous bounded $f$, average $f(Qx)$ over normalized <Haar measure> of the compact <orthogonal group>. Right invariance makes this average depend only on $|x|$; averaging also over <surface measure on a sphere> identifies it with the spherical average. Invariance of $\mu$ and <Fubini's theorem> then give the identity. Uniqueness of finite <Borel measures> and measurable approximation extend it to the stated classes of $f$. No extra sphere-area factor belongs in an average against normalized <surface measure on a sphere>.
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