“Frostman lemma”的意思、由来-开放百科全书

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Frostman lemma

释义

References

{{short description|Tool for estimating the Hausdorff dimension of sets}}

In mathematics, and more specifically, in the theory of fractal dimensions, Frostman's lemma provides a convenient tool for estimating the Hausdorff dimension of sets.

Lemma: Let A be a Borel subset of Rⁿ, and let s > 0. Then the following are equivalent:

H^s(A) > 0, where H^s denotes the s-dimensional Hausdorff measure.
There is an (unsigned) Borel measure μ satisfying μ(A) > 0, and such that

holds for all x ∈ Rⁿ and r>0.

Otto Frostman proved this lemma for closed sets A as part of his PhD dissertation at Lund University in 1935. The generalization to Borel sets is more involved, and requires the theory of Suslin sets.

A useful corollary of Frostman's lemma requires the notions of the s-capacity of a Borel set A ⊂ Rⁿ, which is defined by

(Here, we take inf ∅ = ∞ and {{Frac|1|∞}} = 0. As before, the measure is unsigned.) It follows from Frostman's lemma that for Borel A ⊂ Rⁿ

References

{{Citation

| last1=Mattila
| first1=Pertti
| author1-link = Pertti Mattila
| title=Geometry of sets and measures in Euclidean spaces | publisher=Cambridge University Press
| isbn=978-0-521-65595-8 | year=1995
| mr = 1333890 |series= Cambridge Studies in Advanced Mathematics | volume = 44}}{{mathanalysis-stub}}

3 : Dimension theory|Fractals|Metric geometry

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