A mean value theorem for system of integrals claims that, given any set of continuous functions on I⊆R, and a finite measure μ on I, there exists an n-point quadrature rule which is exact for those functions. If we consider μ as a Gaussian measure on the Borel sigma-field of R, we can use the Hermite polynomials to determine easily the nodes and coefficients of this quadrature rule. The case when measure μ is introduced by a Gaussian stochastic process has been investigated.