Abstract
A Haar measure on a topological group G is a measure on the σ-algebra of the Borel sets of G (that is, the σ-algebra generated by its open subsets), which is invariant under translations in the group. A Haar measure may be left or right invariant. In this chapter, the construction of Haar measures on locally compact topological groups is done. It will also be proved the uniqueness, up to multiplication by a positive constant, of the Haar measure.
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Notes
- 1.
The choice of translations xV leads to a left invariant Haar measure. The same argument applies to translations V x, yielding right invariance.
- 2.
The left invariance of λ V stated in this lemma is a consequence of the choice of covers of type xV . The choice of open sets V x would lead to right invariance.
- 3.
In this argument, the hypothesis that G is Hausdorff is essential to separate the compact sets K 1 and K 2.
- 4.
In this lemma, there appears again the need to work with Hausdorff spaces.
- 5.
The use of prefix “pre” is due to the fact that the set of open sets is not a σ-algebra.
- 6.
See, for instance, Section 11 of Halmos.
- 7.
See, for instance, Section 52 of Halmos [18] and more specifically the F theorem.
References
HALMOS, P. R. Measure theory. Springer-Verlag, 1974.
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San Martin, L.A.B. (2021). Haar Measure. In: Lie Groups. Latin American Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-61824-7_3
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DOI: https://doi.org/10.1007/978-3-030-61824-7_3
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