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위키데이터

• ID : Q1162676

말뭉치

1. Accordingly, one speaks of a left- or right-invariant Haar measure.
2. 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.
3. 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.
4. 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.
5. Then (X, ∑, μ) is called a Haar measure space over X by Fremlin (200?).
6. What makes this into a Haar measure is the fact that it is translation invariant.
7. A Haar measure should be invariant under the group operation.
8. 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.
9. 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.
10. There is an analogy between Haar measure and scaled-cardinality on a finite group.
11. Any locally compact Hausdorff topological group G G admits a Haar integral (and therefore Haar measure) that is unique up to scalar multiple.
12. Some authors define a Haar measure on Baire sets rather than Borel sets.
13. 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.
14. \mu _{A}} extends to a positive linear functional on compactly supported continuous functions and so gives a Haar measure.
15. Von Neumann gave a method of constructing Haar measure using mean values of functions, though it only works for compact groups.
16. In this paper, we prove that the regular left Haar measure in a locally compact topological IP-loop is essentially unique.

메타데이터

위키데이터

• ID : Q1162676

Spacy 패턴 목록

• [{'LOWER': 'haar'}, {'LEMMA': 'measure'}]