Abstract
If G is a locally compact group, there is, up to a constant multiple, a unique regular Borel measure μ L that is invariant under left translation. Here left translation invariance means that μ(X) = μ(gX) for all measurable sets X.
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References
P. Halmos. Measure Theory. D. Van Nostrand Company, Inc., New York, N. Y., 1950.
E. Hewitt and K. Ross. Abstract Harmonic Analysis. Vol. I, Structure of topological groups, integration theory, group representations, volume 115 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1979.
L. Loomis. An Introduction to Abstract Harmonic Analysis. D. Van Nostrand Company, Inc., Toronto-New York-London, 1953.
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Bump, D. (2013). Haar Measure. In: Lie Groups. Graduate Texts in Mathematics, vol 225. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8024-2_1
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DOI: https://doi.org/10.1007/978-1-4614-8024-2_1
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