ABSTRACT
Lebesgue measure on R $ \mathbb R $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315151601/40b40f18-f94e-4633-83f0-b519efc5b015/content/inline-math16_1.tif"/> and counting measure on Z $ \mathbb Z $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315151601/40b40f18-f94e-4633-83f0-b519efc5b015/content/inline-math16_2.tif"/> are examples of measures μ $ \mu $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315151601/40b40f18-f94e-4633-83f0-b519efc5b015/content/inline-math16_3.tif"/> that are translation invariant, that is, μ ( B + x ) = μ ( B ) $ \mu (B + \boldsymbol{x}) = \mu (B) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315151601/40b40f18-f94e-4633-83f0-b519efc5b015/content/inline-math16_4.tif"/> for all Borel sets B. These are special cases of a general construct called Haar measure. As we shall see, the existence Haar measure leads to a unification and generalization of Fourier analysis, the basic aspects of which are presented in this chapter.
