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.