Radial disintegration of an orthogonally invariant measure ($\int fd\mu ={\int }_{[0,\infty )}{\int }_{{\mathbb{S}}^{d-1}}f(r\theta )d\sigma (\theta )d(|\cdot {|}_{\#}\mu )(r)$)

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.