Diagonals of real symmetric matrices of given spectra as a measure space
The set of diagonals of real symmetric matrices of given non negative spectrum is endowed with a measure which is obtained by the push forward of the Haar measure of the real orthogonal group.\\ We prove that the Radon Nicodym derivation of this measure with respect to the relative Euclidean measure is approximated by the coefficients of a sequence of zonal sphere polynomials corresponding with the given spectrum. 
There is a striking similarity between the role of the zonal sphere polynomials in the orthogonal case, and that of the Schur function in the Hermitian case.\\ Following this we obtain a combinatorial approximation for the probability of real symmetric matrix of a given spectrum to appear as the sum of two real symmetric matrices, each of a given spectrum. In addition we obtain a real orthogonal analogue to the Zuber Itzykson Harish Chandra integration formula.