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Open‐Invariant Measures and the Covering Number of Sets
Author(s) -
Haase Hermann
Publication year - 1987
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19871340121
Subject(s) - mathematics , open set , invariant (physics) , compact space , measure (data warehouse) , invariant measure , metric space , hausdorff space , locally compact space , borel measure , hausdorff measure , discrete mathematics , pure mathematics , combinatorics , probability measure , hausdorff dimension , data mining , computer science , ergodic theory , mathematical physics
A result of J. Mycielski says that on every metric space ( X , ϱ) with a non‐empty compact thick set C ⫅ X there exists a regular open‐invariant Borel measure μ with μ (C) = 1. μ is called open‐invariant if μ (A) = μ (B) for open isometric sets A , B ⫅ X . We relate this result to the notion of a Hewitt‐Stromberg measure and give a new independent existence proof for such an open‐invariant measure μ on a compact metric space ( X , ϱ). This proof works by induction, the well‐known metric outer measure construction of Caratheodory‐Hausdorff and a new property of the covering number N(X, q) of X .
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