Densities on locally compact abelian groups
نویسنده
چکیده
A density on a locally compact Abelian group G is a bounded system of compatible measures on the compact quotients of G. The notion of density arises naturally both in connection with the theory of uniform distribution and with the theory of almost periodic functions. Our main result effectively answers the question "when are all densities on G induced by measures on some compactification of G?" This turns out to be equivalent to the question "when does each density on G extend to a bounded linear functional on the space of semiperiodic functions on G?" Let G be a locally compact Abelian (LCA) group. DEFINITION. A density /j, on the LCA group G is a system of measures on the compact quotients of G constructed as follows. To each compact quotient Q of G is associated a regular Borel measure IXQ defined on the Borel sets of Q and satisfying (1) If E is a Borel set of Q and if Q is a quotient of Q then JUQ(E) = MQ'(0 1 C£))> where 0 : Q-^Q is the natural homomorphism.
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