Let n ∈ N where set A ⊆ Rn. Suppose a set A is unbounded, if for any r>0 and x0 ∈ Rn, d(x,x0)>r for some x ∈ A, where d is the standard Euclidean Metric of Rn.
If U is the set of all unbounded A measurable in the Caratheodory sense, using the Hausodorff Outer measure (pg. 2, def. 2 of paper), and the mean of A is taken w.r.t the Hausdorff measure and dimension (pg. 2-3, def. 3-4), then prove:
The mean of A is finite, for a subset of U with a cardinality only less than |U|.
Juan Weisz A power set of real numbers is said to have a cardinality.
https://math.stackexchange.com/questions/93991/what-is-known-about-the-power-set-of-the-real-number-line
The power set is the sets of all subsets of R. Further, the power set of Rn is a superset of the sets of all unbounded subsets of Rn or U.
Juan Weisz Mean of A, a subset of Rn. (How do I make my post more clear?)
Juan Weisz In the attachment (skip the abstract and intro) the paper shows how to find the mean of A, using the Hausdorff measure. See def. 3-6. (Note in def. 5, function “f:A->R” should be just set “A”).
If you say your sete are unbounded.
I cannot see a finite mean in tthe sense
Of my understanding.
Of course if you mention housdorf, you can
Be in a situation of reduced dimensionality
Or an intermediate one.
If you’re correct that no unbounded set has a finite mean, then that already proves the statement.…
I should probably state I want to extend the mean of A (see the attachment) such that the mean is finite for a subset of U (def. 5 of paper) with the same cardinality as that of U.
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