I am trying to understand in which metric spaces the metric is jointly measurable.

There exist a metric space $(X,d)$ for which the Borel $\sigma$-algebra, does not coincide with the product Borel $\sigma$-algebra, that is $\mathcal{B}(X)\otimes \mathcal{B}(X) \subsetneq B(X\times X)$. Every construction, I have encountered, of such a (non-separable) metric space one considers a metric space of cardinality strictly greater than the continuum $\mathfrak{c}$. For in such a metric space the diagonal $I=\{(x,x):x\in X\}\in B(X\times X)$, is not an element of $\mathcal{B}(X)\otimes \mathcal{B}(X)$ (c.f. Nedoma's pathology). This also proves that $d$ is not jointly measurable, since $d^{-1}(\{0\})=I\not\in\mathcal{B}(X)\otimes \mathcal{B}(X)$.

For metric spaces $(X,d)$ with $\text{card}(X)\leq \aleph_0$, we obviously have that $d$ is jointly measurable, but the area of interest is : What happends when the cardinality is $\aleph_0 < \text{card}(X) \leq \mathfrak{c}$?

More specifically, I have been wondering about the following questions: Let $(X,d)$ be a metric space,

- Is the metric $d$ jointly-measurable, if $\aleph_0 < \text{card}(X) \leq \mathfrak{c}$?
- If not, is there equivalence between joint measurability of $d$ and separability of $X$?
- If not, do you know property of $(X,d)$, weaker than separability, that ensures joint-measurability of $d$?