How are joint probability distributions constructed from product measures? I often see a construction in measure theory in regards to product measures. This is outlined below (taken from Wikipedia because it's very generic)

Let $ (X_{1},\Sigma _{1})$ and $(X_{2},\Sigma _{2})$ be two measurable
  spaces, that is, $\Sigma _{1} $ and $\Sigma _{2}$ are sigma algebras
  on $X_{1}$ and $X_{2}$ respectively, and let $\mu _{1} $and $\mu _{2}$
  be measures on these spaces. Denote by $\Sigma _{1}\otimes \Sigma_{2}$ the sigma algebra > on the Cartesian product $X_{1}\times X_{2} $ generated by subsets of the form $B_{1}\times B_{2}$, where $B_{1}\in
> \Sigma _{1}$ and $B_{2}\in \Sigma _{2}.$ This sigma algebra is called
  the tensor-product σ-algebra on the product space. A product measure
  $\mu _{1}\times \mu _{2}$ is defined to be a measure on the measurable
  space $(X_{1}\times X_{2},\Sigma _{1}\otimes \Sigma _{2})$ satisfying
  the property $(\mu _{1}\times \mu _{2})(B_{1}\times B_{2})=\mu
> _{1}(B_{1})\mu _{2}(B_{2})$ for all $B_{1}\in \Sigma _{1},\ B_{2}\in \Sigma _{2}.$

Questions:


*

*From the perspective of probability theory, this product measure construction looks  a lot like the construction of a joint probability $P(X,Y)$ where the $X$ and $Y$ are independent. Is this assumption correct?

*If (1.) is correct, then how does the notion of correlation come into the structure of product measures? How is correlation built into measure theory so that it can pass onto probability theory?
 A: *

*

You're correct that the statement of independence is, that if $X$ and $Y$ are (real-valued) random variables on a common probability space with respective distributions $X(P)$ and $Y(P)$, then $X$ and $Y$ are said to be independent exactly if the distribution of $(X,Y)$ (which is a random vector and thus, its distribution is a measure on $\mathbb{R}^2$) is the product measure $X(P)\otimes Y(P)$.
2.
Correlation is not inherent to product measures since these are exactly the distributions of independent random variables - in particular, uncorrelated ones. However, when $X$ and $Y$ are not independent, then $(X,Y)(P)$ is not a product measure, but some other measure on $\mathbb{R}^2$. This allows for the covariance
$$
\int_{\mathbb{R}^2} xy\; \textrm{d}(X,Y)(P)-\int_{\mathbb{R}^2} xy\;\textrm{d}X(P)\otimes Y(P)
$$
to be non-trivial.
3.
About the relation between product measures and other measures.
One way that general measures do pop up from product measures is via conditional distributions (see, for instance, https://en.wikipedia.org/wiki/Regular_conditional_probability). If you have a good notion that "conditional on $Y=y$, then $X$ follows the distribution $\mu_y$," then you have a scheme as follows: Let $(U,Y)$ be an independent pair such that $Y$ is still $Y$ and $U$ is uniform on $[0,1]$. Then, if $F_y$ is the cumulative distribution function of $\mu_y$ with right-continuous generalised inverse $F_y^{-1}$, we have that $F_y^{-1}(U)$ follows the distribution $\mu_y$.
Therefore, given such a pair $(U,Y)$, the transformation $(U,Y)\mapsto (F_Y(U),Y)$ yields a variable with the same distribution as $(X,Y)$. Thus, the latter distribution (which is not a product measure) can be constructed from the former (which is).
