Direct proof of existence of the tensorial moment measure μ_T

Establish a direct proof that, for any real symmetric tensor T, the sequence of moments m_n(T) defined via tensor trace invariants is the moment sequence of a probability measure μ_T on the real line; that is, prove directly that there exists a probability measure μ_T on R such that for all n ≥ 0, ∫ λ^n dμ_T(λ) = m_n(T).

Background

The paper defines moments m_n(T) of a tensor T as sums over connected rooted p-regular trace maps and cites a result of Gurau asserting the existence of a probability measure μ_T with these moments. However, the authors point out that a direct proof of this existence is not currently available.

A direct proof would strengthen the theoretical foundations of the tensorial moment framework by providing an intrinsic argument (independent of external constructions) that the moment sequence associated with a real symmetric tensor indeed corresponds to a probability measure.