Abstract
Chapter 1 contains a proof that Lebesgue measure is translation invariant. 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. Chapter 9 is devoted to an introduction to Haar measure.
Section 9.1 contains some basic definitions and facts about topological groups. 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. In Section 9.4 we construct two algebras that are fundamental for the study of harmonic analysis on a locally compact group.
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Notes
- 1.
Recall that a square matrix with real entries is orthogonal if the product of it with its transpose is the identity matrix.
- 2.
A map \(F : {\mathbb{R}}^{d} \rightarrow {\mathbb{R}}^{d}\) is affine if there exist a linear map \(G: {\mathbb{R}}^{d} \rightarrow {\mathbb{R}}^{d}\) and an element b of \({\mathbb{R}}^{d}\) such that \(F(x) = G(x) + b\) holds for each x in \({\mathbb{R}}^{d}\). If F is affine, then G and b are uniquely determined by F, and we will (for simplicity) denote by det(F) the determinant of the linear part G of F (see Sect. 6.1).
- 3.
In particular, the reader who is interested only in second countable locally compact groups can ignore the references to Sect. 7.6 in what follows.
- 4.
The function g cannot be defined simply be requiring that its restriction to each C in \(\mathcal{H}\) be g C ; see Exercise 12.
- 5.
Recall that a directed set is a partially ordered set A (say ordered by ≤ ) such that for each α and β in A, there is an element γ of A that satisfies α ≤ γ and β ≤ γ. A net is a family indexed by a directed set. A net \(\{x_{\alpha }\}_{\alpha \in A}\) in a topological space X is said to converge to a point x of X if for each open neighborhood U of x there is an element α 0 of A such that x α ∈ U holds whenever α satisfies α ≥ α 0. Thus \(\lim _{\alpha }\|f {\ast}\varphi _{\alpha }- f\|_{1} = 0\) holds if and only if for each positive ε there is an element α 0 of A such that \(\|f {\ast}\varphi _{\alpha }- f\|_{1} <\epsilon\) holds whenever α satisfies α ≥ α 0. See Kelley [69] for an extended treatment of nets.
References
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Cohn, D.L. (2013). Haar Measure. In: Measure Theory. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6956-8_9
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