# Existence proof for two random variables

I have two probability measures $\nu_1,\nu_2$ on a measurable set $(E,\Sigma)$ and a probability measure $\mu$ on $(E \times E, \sigma(\Sigma \times \Sigma))$ with $$\nu_1(A) = \mu(A \times E) \quad \nu_2(A) = \mu( E \times A)$$ for all $A\in \Sigma$. I would like to claim that there exist two random variables $X_i : \Omega \to E, \,i=1,2$ with $$\mu(A \times B) = \mathbb{P}(X_1 \in A , \, X_2 \in B).$$ I found that this follows from the Daniell-Kolmogorov theorem on a polish space in the book probability thery from Bauer, Proposition 35.3. However, in this theorem one proofs it for an arbitary set of measures and not of just of simply 2. Can I drop here the requirement of a polish space and proof the existence of two such random variables?

Of course, it does: consider $\Omega$ being the product space and projections being the random variables.