Let $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ be two measurable spaces and let $(\mathbf{P}_n)_{n=1}^{\infty}$ be a sequence of probability measures in $(X\times Y, \mathcal{A} \otimes \mathcal{B})$ that satisfies the following condition:
There exists a set function $\mathbf{P}\colon \mathcal{A} \times \mathcal{B}\rightarrow \mathbb{R}$ such that for every $A\times B \in \mathcal{A} \times \mathcal{B}$ it holds that $\mathbf{P}_n(A\times B)\rightarrow \mathbf{P}(A\times B)$, when $n\rightarrow \infty$.
The question is, can $\mathbf{P}$ always be extended to a measure $\hat{\mathbf{P}}$ in $(X\times Y, \mathcal{A} \otimes \mathcal{B})$ or are there counterexamples? Furthermore, is this extension unique if it exists?