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- ID : Q1162676
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- Accordingly, one speaks of a left- or right-invariant Haar measure.[1]
- The concrete examples described below provide a direct connection between the rather abstract theory of Haar measure and its application to situations which are relevan't in statistical applications.[2]
- It turns out that very similar results hold for every locally compact group (see Section 9.1 for the definition of such groups); the role of Lebesgue measure is played by what is called Haar measure.[3]
- Section 9.2 contains a proof of the existence and uniqueness of Haar measure, and Section 9.3 contains additional basic properties of Haar measures.[3]
- Then (X, ∑, μ) is called a Haar measure space over X by Fremlin (200?).[4]
- What makes this into a Haar measure is the fact that it is translation invariant.[5]
- A Haar measure should be invariant under the group operation.[5]
- In 1937, Stefen Banach wrote an appendix about the Haar measure for a textbook about integration theory, and in 1940 both Weil and Cartan offered new uniqueness proofs.[6]
- However, because the Haar measure is more general and abstract, it can illuminate and even justify the choice of the Lebesgue measure as the natural measure.[6]
- There is an analogy between Haar measure and scaled-cardinality on a finite group.[7]
- Any locally compact Hausdorff topological group G G admits a Haar integral (and therefore Haar measure) that is unique up to scalar multiple.[7]
- Some authors define a Haar measure on Baire sets rather than Borel sets.[8]
- The left Haar measure satisfies the inner regularity condition for all σ {\displaystyle \sigma } -finite Borel sets, but may not be inner regular for all Borel sets.[8]
- \mu _{A}} extends to a positive linear functional on compactly supported continuous functions and so gives a Haar measure.[8]
- Von Neumann gave a method of constructing Haar measure using mean values of functions, though it only works for compact groups.[8]
- In this paper, we prove that the regular left Haar measure in a locally compact topological IP-loop is essentially unique.[9]
소스
- ↑ Encyclopedia of Mathematics
- ↑ Easton: Chapter 1: Integrals and the Haar measure
- ↑ 3.0 3.1 Haar Measure
- ↑ Haar Measure - an overview
- ↑ 5.0 5.1 What is Haar Measure?
- ↑ 6.0 6.1 The Concept of Probability in Statistical Physics
- ↑ 7.0 7.1 Haar integral in nLab
- ↑ 8.0 8.1 8.2 8.3 Haar measure
- ↑ The uniquenesss of a left Haar measure in topological IP-loops
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위키데이터
- ID : Q1162676