  The moment problem is an important problem in Functional Analysis and in
Probability measure. It goes back to Stieltjes, around 1890. There is still an
important ongoing interest in the recent literature. But, up today, the main
theoretical resource (Shohat and Tamarkin, 1934) does not have the modern
exposure it deserves, especially in the current development of measure theory
of integration. Besides, the multivariate version is far less exploited. In
this paper, a full exposure of such a theory is presented, using the latest
knowledge of measure theory and functional analysis. As a result, the basis of
future development is layed out and the accessibility of the theory by modern
graduate students andresearches is guaranteed.
