Measures on product spaces

Purpose of the present paper is the study of probability measures on countable\footnote{ All the results still hold without this hypothesis, which is done to keep evident the analogy with states of uniformly hyperfinite algebras (cfr. [6], [10]).} products of measurable spaces. We will discuss two problems: 1) equivalence of measures on product spaces; classification\footnote{For example $(\Psi_k)_{k\in\mathbb N}$ are always cylindrical measures when $\Psi_k= \prod\limits^k_{i=1}m_i$ ($m_i$, a measure on $(\Omega_i;{\cal B}_i)$) or when $(\Omega_i,{\cal B}_i)=(\Omega,{\cal B})$ for every $\iota\in\mathbb N$ and the $\Psi_k$'s are the measures induced by a Markov chain (this follows from Ionesco Tulcea's theorem; (cfr. [5], pg. 162). Or, finally, because of Kolmogorov's extension theorem, when the $(\Omega_i;{\cal B}_i)$ are standard Borel space (cfr. [5], pg. 83).} of measures on such spaces.