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analysis

Source: miklos schweitzer 1998 q4

September 18, 2021
real analysisMeasure theory

Problem Statement

For any measurable set H⊂RH \subset R , we define the sequence an(H)a_n(H) by the formula: an(H)=λ([0,1]∖⋃k=n2n(H+log⁡2k))a_n(H) = \lambda \bigg([0,1] \setminus \bigcup_{k = n}^{2n} (H + \log_2 k) \bigg) where λ\lambda denotes the Lebesgue measure and log⁡2\log_2 denotes the binary logarithm. Prove that there is a measurable, 1-periodic, positive measure set H⊂RH \subset R , such that the sequence an(H)a_n( H ) does not belong to any space lpl_p (1≤p<∞1 \leq p < \infty).
[hide=not sure about this part]For what numbers 1≤p<∞1 \leq p <\infty is it true that whenever H is 1-periodic, positive measure, the sequence an(H)a_n( H ) belongs to the space lpl_p?
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