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INTEGRATION ON LOCALLY COMPACT SPACES 1 Introduction
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affine transformations arbitrary Cartesian product closed subset commutator subgroup compact neighborhood compact subset compact support contained continuous and open continuous homomorphism Corollary denote det(T disjoint equal Example exists extended-real-valued functions defined finite number follows given group G Hausdorff space homomorphism of G if(K inner automorphisms integral on G integration with respect invariant Haar integral invariant integrals invariant positive integral Jf(E Jf(G Jf+(G Lebesgue integral left invariant Haar Lemma linear functional locally compact group locally compact space lower semicontinuous measure modulus morphism necessary and sufficient null function obtain open covering open homomorphism open subset operates topologically proof Proposition 13 proved Q.EJ Q.EJX real number real-valued continuous function real-valued function relatively invariant positive right invariant Haar scalar product Section subgroup of G theorem topological automorphism topological group topological homogeneous space topological space topological subgroup topologically isomorphic unique vector subspace whence