Let $(X,d_X),(Y,d_Y)$ be metric spaces and $X$ is locally-compact and fix a Borel probability measure $\nu$ on $X$. For any Borel-measurable $f:X\rightarrow Y$, let $\mathcal{K}(f,\delta)$ be the set of compact subsets $K$ of $X$ on which $f|_K$ is continuous and for which $\nu(X-K)<\delta$.
Consider the following metrics on the set of Borel functions from $X$ to $Y$. $$ \begin{aligned} d^+(f,g):= &\operatorname{argmin}\left\{ \delta>0:\, \sup_{K\in \mathcal{K}(f,\delta) \cap\mathcal{K}(g,\delta)} \sup_{x \in K} d_Y(f(x),g(x))\leq \delta \right\}.\\ d^-(f,g):= &\operatorname{argmin}\left\{ \delta>0:\, \inf_{K\in \mathcal{K}(f,\delta) \cap\mathcal{K}(g,\delta)} \sup_{x \in K} d_Y(f(x),g(x))\leq \delta \right\}. \end{aligned} $$ Are my of these studied in the literature? If so, what are they called and which topologies do they metrize?
This seems to be some sort of uniform variant of the Ky-Fan metric...
