In this article, we considered (1.1) a family of proper functions H = (fᵢ) defined in a locally compact-space E and separating its points. The moments of a measure μ on E relative to the proper injection f = (fᵢ) are defined as the integrals μ (eₚ) where eₚ belongs to the canonical base of the algebra of polynomials of real coefficients with regard to the family H. In propositions (1): we have shown that if the space E is compact, any measure  on E is uniquely defined by the moments μ (eₚ) . In proposition (2), we have shown that if E is a locally compact space which is not compact, the integrals μ (eₚ) of the proplonged monomials eₚ, define, uniquely, a bounded measure μ on E. Some relation may be found in 1.4 between the result of proposition (2) and the theorem of P.Levy on characteristic functions. A particular application of proposition (2) in the field of applications of the theory of probability may be also found in 1.5.