Suppose $(X,\mathcal{B},\mu)$ is a measure space, and let $B\subseteq X^2$ be an arbitrary set.

1) Is there a nice characterization of the circumstances under which there is a $\sigma$-algebra $\mathcal{C}\supseteq\mathcal{B}$ and a measure $\nu$ on $\mathcal{C}$ extending $\mu$ such that $B$ is measurable with respect to $\mathcal{C}\times\mathcal{C}$?

2) Is it possible for there to be distinct extensions $\mathcal{C}_1,\nu_1$ and $\mathcal{C}_2,\nu_2$ such that $B$ is measurable with respect to $\mathcal{C}_1\times\mathcal{C}_1$ and $\mathcal{C}_2\times\mathcal{C}_2$, but $\nu_1^2(B)\neq \nu_2^2(B)$?

3) If the answer to 2 is yes, is there a nice characterization of the circumstances under which extensions of $\mathcal{B}$ assign a unique value to the measure of $B$?

(If it helps, $\mathcal{B}$ can be assumed to have or not have nice properties like separability or being a regular Borel measure.)