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I learned from this paper about the Verblunsky theorem. My question is that: What kind of generalizations of this theorem is availlable? In particular I am interested in the following two possible ...

Let $X$ be a compact metric space, and $\mu$ a Borel probability measure. For $S\subset\mathbb{N}$ we denote the upper density with $\overline{D}(S).$ Let $f:X\rightarrow2^{\mathbb{N}}$ be a ...