Orthogonal probability representation of spherical averages

There is one probability distribution on the closed operator-norm unit ball of linear endomorphisms of $\mathbb R^3$, concentrated on orthogonal transformations, with the identity transformation in its topological support, such that for every continuous function $g$ and every $x\in\mathbb R^3$,

Here $\overline g$ is the normalized surface average on the sphere of radius $\lVert x\rVert$, including its value $g(0)$ at the origin. This is the compact-group averaging representation in equation (77) of the source, stated using continuous linear operators to give the parameter space its standard finite-dimensional topology. Normalized Haar measure on the orthogonal group, pushed into the operator ball, gives the required distribution. No differentiability, field equation, positivity or behavior at infinity is assumed.

This helper is used by the Hessian-barycenter reduction: the reduction proves smooth differentiation under the compact average, orthogonal invariance of the trace and Hessian norm, the Hessian moments and the support property. Only the measure-theoretic representation of the spherical average remains here.