Coordinatewise product-moment functional for tensor polynomials

For $d\ge 1$, let $(a_j,b_j,m_j,2m_j-1,{\mu }_j)$ be Gaussian moment data for each coordinate $j=1,\dots ,d$, and write $M_k^{(j)}:=\int x^{k}d{\mu }_j(x)$. A coordinatewise tensor polynomial is a finite family $(c_{\alpha })$, with $c_{\alpha }\in \mathbb{R}$ for every index, indexed by multi-indices $\alpha =({\alpha }_1,\dots ,{\alpha }_d)$ with $0\le {\alpha }_j\le 2m_j-1$, evaluated by $P(x_1,\dots ,x_d)={\sum }_{\alpha }{c}_{\alpha }{\prod }_{j=1}^d{x}_j^{{\alpha }_j}\mathrm{.}$ Define its coordinatewise product-moment value by $I_{\otimes }(P):={\sum }_{\alpha }{c}_{\alpha }{\prod }_{j=1}^d{M}_{{\alpha }_j}^{(j)}\mathrm{.}$ In both sums, list the finite multi-index box in increasing lexicographic order (coordinate $1$ first, then coordinate $2$, and so on), and take the ordinary finite-list sum in that order. The products use increasing coordinate order $j=1,\dots ,d$. The real coefficient family is part of the formal polynomial data, so these formulas specify finite real sums. This definition specifies the product-moment functional used for coordinatewise exactness without requiring a measure-product construction.