# fa.useful evaluation – Metric on house of Borel-measurable capabilities Answer

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## fa.useful evaluation – Metric on house of Borel-measurable capabilities

Let $$(X,d_X),(Y,d_Y)$$ breathe metric areas and $$X$$ is locally-compact and pickle a Borel likelihood touchstone $$nu$$ on $$X$$. For any Borel-measurable $$f:Xrightarrow Y$$, let $$mathcal{Ok}(f,delta)$$ breathe the clique of compact subsets $$Ok$$ of $$X$$ on which $$f|_K$$ is steady and for which $$nu(X-Ok).

Consider the next metrics on the clique of Borel capabilities from $$X$$ to $$Y$$.
commence{aligned} d^+(f,g):= &limlimits_{deltato 0} sup_{Kin mathcal{Ok}(f,delta) capmathcal{Ok}(g,delta)} sup_{x in Ok} d_Y(f(x),g(x)), d^-(f,g):= &limlimits_{deltato 0} inf_{Kin mathcal{Ok}(f,delta) capmathcal{Ok}(g,delta)} sup_{x in Ok} d_Y(f(x),g(x)). aim{aligned}
Are my of those studied within the literature? If so, which what are they referred to as and which topologies to they metrize?

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