Consider the following excerpt from "Distance covariance in metric spaces" by Ryssell Lyons:

enter image description here Both in the Lemma and in order to define $D(\mu)$ one needs that $d:X\times X \to [0,\infty]$ is $ \mathcal{B}(X) \otimes \mathcal{B}(X) /\mathcal{B}(\mathbb{R})$ measurable, but in the proof of the lemma this is omitted.

In the "Handbook of Analysis and its Foundations" by Eric Schechter, one finds the following corrollary to Nedoma's Pathology enter image description here

From this we can deduce, that the Russell Lyons article is erroneous in that the described setup (with $X$ being a general metric space) allows for integrals that are not well-defined.

We know that if $X$ is separable then $d$ is jointly measurable, but is there a (easily comprehensible) property of metric spaces, which is equivalent to saying that the metric $d$ is $\mathcal{B}(X) \otimes \mathcal{B}(X)$-measurable?



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