Assuming countable and dependent choice, countable products of arbitrary probability spaces

Define ${\mu }_0({\pi }_F^{-1}(A))=({⨂}_{n\in F}{\mu }_n)(A)$. The finite product marginals are consistent, so [F1] applies.

By [F2], ${\mu }_0$ extends to a measure $\mu$ on the generated cylinder sigma-algebra. Since the empty cylinder is $E$ and has value $1$, $\mu$ is a probability measure.

Let $\nu$ be another probability measure with the stated marginals. The family $\mathcal{D}=\{A\in {\mathcal{C}}_{\mathbb{N}}:\mu (A)=\nu (A)\}$ is a lambda-system: it contains the whole space because both measures have mass one, is closed under relative complements of nested members, and is closed under increasing countable unions by continuity from below. The two measures agree on every cylinder, and cylinders are a pi-system by Finite-coordinate cylinders form a $\pi$-system, so [F3] gives ${\mathcal{C}}_{\mathbb{N}}\subseteq \mathcal{D}$. Hence $\mu =\nu$.