Suppose we have two random variables $X_i$ on the probability space $(\Omega_i, \sigma(\Omega_i),P_i)\; i = 1,2$. Now, $P_i$ is just a technical tool and we consider directly the distribution of $X_i$ as a measure in the new probability spaces : \begin{align} (\mathbb{R}, \mathcal{B}(\mathbb{R}), P_{X_i}) \end{align}
My knowledge is that we can construct a product space by considering $X_i$ independents. The new measure is then the product measure : \begin{align} \lambda = P_{X_1} \otimes P_{X_2} \end{align}

I also know that if we have the space $(\mathbb{R}^2, \mathcal{B}(\mathbb{R}^2), P)$, the existence of regular conditional probability tells us that : \begin{align} P(A\times B) = \int_B p(A|\omega)dP_1 = \int_A p(B|\omega)dP_2 \end{align} Where $P_i\; i = 1,2$ are the marginal measure of $P$.

So if the space is the product space generated by $\lambda = P_{X_1} \otimes P_{X_2}$ we have : \begin{align} P(A\times B) =& \int_B p(A|\omega)dP_{X_1} = \\ =& \int_B P_{X_2}(A)dP_{X_1} \\ =& P_{X_2}(A)\int_B dP_{X_1} \\ =& P_{X_2}(A)P_{X_1}(B) \end{align}

My questions are :

1) if $P_{X_i}$ are not independent, how goes the construction of the product measure ?

2) Can we construct a probability space where $P_{X_i}$ are marginals of a special measure retaining the fact that they are dependent ?

3) More specifically, I want to know how to construct a product probability space starting from some measures in a general way. Without knowing if the measure are independent or not.


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