Finding a choice function which chooses a unique set of equivalent sequences of sets, where the expected value is unique, natural and finite?

03 July 2023 1 771 Report

(See attached article for more info).

Let (X,d) be a metric space. If set A⊆X, let Hα be the α-dimensional Hausdorff measure on A, where α∈[0,+∞) and dimH(A) is the Hausdorff measure of set A.

In addition, when n∈N, with set A⊆Rn and function f:A→R; consider the following definitions:

If we define a sequence of sets where (Fr*)r∈N, where if h is the dimension function (https://en.wikipedia.org/wiki/Dimension_function), then when:

the set theoretic limit (https://en.wikipedia.org/wiki/Set-theoretic_limit) of (Fr*)r∈N is {(x,f(x)):x∈A} (i.e. (Fr*)r∈N converges to {(x,f(x)):x∈A})

For all r∈N, 0

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