Math 2.0

Potential theory Definition

Sets of capacity zero

Builds on Compact setLogarithmic capacityCapacity of a setPolar set

Definition

Let n≥2n \ge 2 and let E⊆ℝnE \subseteq \R^n. For n=2n = 2, identify ℝ2\R^2 with ℂ\C and let cap⁡\operatorname{cap} be the logarithmic capacity. For n≥3n \ge 3, let cap⁡\operatorname{cap} be the Newtonian capacity. The set EE has capacity zero if cap⁡(K)=0\operatorname{cap}(K) = 0 for every compact set K⊆EK \subseteq E.

Remarks

This is the condition that the inner capacity of EE vanishes. For Borel sets it is equivalent to the vanishing of the outer capacity, by Choquet's capacitability theorem.

Example: in ℂ\C, a finite set KK has capacity zero, since every probability measure on KK has an atom and so has infinite logarithmic energy. Every compact subset of a countable set is countable, and a countable compact set carries no atomless probability measure, so countable subsets of ℂ\C also have capacity zero.

Non-example: the segment [0,1][0, 1] in ℂ\C does not have capacity zero, since it is compact with logarithmic capacity 1/41/4.

A Borel set has capacity zero if and only if it is a polar set. In ℝn\R^n, n≥3n \ge 3, a compact set with finite (n−2)(n-2)-dimensional Hausdorff measure has capacity zero.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Capacity of a set, concept, no evidence levelCapacity of a setconceptCompact set, concept, no evidence levelCompact setconceptLogarithmic capacity, concept, no evidence levelLogarithmic capacityconceptPolar set, concept, no evidence levelPolar setconceptSets of capacity zero, definition, no evidence levelSets of capacity zerodefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Capacity of a set concept
  2. Compact set concept
  3. Logarithmic capacity concept
  4. Polar set concept
  5. Sets of capacity zero definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Capacity_of_a_set
About this page
TagMVYTPJ3V05
URIm2://MVYTPJ3V05
Kinddefinition
WritingW0 stub
MSC31B15, 31A15, 31C15
DefinesSets of capacity zero
UsesCompact set, Logarithmic capacity, Capacity of a set, Polar set

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