A measurable function is exactly one whose dyadic Hausdorff-content Choquet averages recover it at almost every point, and continuous functions are not dense in the corresponding spaces for content dimension delta < n.
R.: A note on Riesz potentials, Duke Math
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On Lebesgue points and measurability with Choquet integrals
A measurable function is exactly one whose dyadic Hausdorff-content Choquet averages recover it at almost every point, and continuous functions are not dense in the corresponding spaces for content dimension delta < n.