Automatic continuity via analytic thinning

Abstract:

We use Choquet’s analytic capacitability theorem and the Kestelman-Borwein-Ditor theorem (on the inclusion of null sequences by translation) to derive results on ‘analytic automaticity’ – for instance, a stronger common generalization of the Jones/Kominek theorems that an additive function whose restriction is continuous/bounded on an analytic set $T$ spanning $\mathbb {R}$ (e.g., containing a Hamel basis) is continuous on $\mathbb {R}$. We obtain results on ‘compact spannability’ – the ability of compact sets to span $\mathbb {R}$. From this, we derive Jones’ Theorem from Kominek’s. We cite several applications, including the Uniform Convergence Theorem of regular variation.