2.4 Measure Theory 23 Haar measure Given a Lie group G, Haar’s theorem says that there is a unique (up to a positive multiplicative constant) nontrivial measure mon the Borel subsets of Gsuch that 1. mis left-translation invariant, i.e. m(gA)=m(A) for all g ∈Gand all Borel sets A⊆G; 2. mis finite on every compact set, i.e. m(K) <∞for all compact K⊆G; 3. mis outer regular on Borel sets A⊆G, i.e. m(A)= infA⊆U,U open m(U); 4. mis inner regular on open sets U ⊆G, i.e. m(U)=supK⊆U,Kcompact m(K). Such a measure monGis called a left-invariant Haar measure. From the above properties, it follows that m(U) > 0 for all non-empty open subsets U ⊆ G. In particular, if Gis compact then m(G) is finite and positive, which allows to specify a unique left-invariant Haar measure on Gby normalising such that m(G)=1. Analogously, there is also a unique right-invariant Haar measure onG, which need not coincide with the left-invariant Haar measure. A Haar measure that is both left-invariant and right-invariant is called a bi-invariant Haar measure. The Haar measure assigns an invariant volume to subsets of Lie groups, which allows to define an integral for functions on such groups. Intuitively, the Haar measure acts a kind of uniform distribution over the group.
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