WebJan 1, 2010 · It is shown that there are two compact sets X,Y whose Hausdorff dimensions are both in (0,1), but all doubling measures on X are purely atomic, while any doubling … WebNov 15, 2024 · We say that a purely atomic finite measure is interval filling if its range is an interval. A sequence of values of such a measure on its atoms is called an interval filling sequence. This notion was introduced in [9] and intensively studied f.e. in [10], [11] and in many other papers. By the Kakeya Theorem, a nonincreasing, summable sequence ...

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WebMay 1, 2011 · Let (S, F, P) be a purely atomic probability space with σ-finite probability measure P then for each event E of F where T is a set of non-equivalent atoms of E. AT WebJan 2, 2024 · Every purely atomic measure is an atomic measure. I am trying to find an example of an atomic measure that is not purely atomic, can anyone help me? measure … jesko treiber

Recovering a purely atomic finite measure from its range

Web$\begingroup$ I don't see why what you said implies there are at most countably many atoms. I can build continuum-many subsets of $[0,1]$ of measure $1/2$ whose pairwise … Webthe set has measure zero.. If is an atom, all the subsets in the -equivalence class [] of are atoms, and [] is called an atomic class. If is a -finite measure, there are countably many … WebSep 15, 2007 · Step 2. We prove that all doubling measures on X are purely atomic. Let µ be a c-doubling measure on X. One has µ { 1 2 }=0 because 1 2 is an accumulation point of X.Toshowµ is purely atomic, i.e. to show µ is supported by uniontext ∞ n=1 S n \ { 1 2 }, it is sufficient to prove µ (E n ) = 0 for any n. lamp 66340