Vol. 23, No. 2, 1967

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A class of measures on the Bohr group

Mahendra Ganpatrao Nadkarni

Vol. 23 (1967), No. 2, 321–328

Let Rd denote the real line with the discrete topology. Let B = Rd be its dual. R, the real line is continuously isomorphic to a dense subgroup of B. Let μ be a finite positive measure defined for Borel subsets of B. Let χt denote the character on B corresponding to the real number t. We shall denote by HS the subspace of L2(B,μ) spanned by {χt : t S}. Assume that −∞<S<HS = {0}. In this case the subspaces HS are strictly increasing in the sense that HS HS whenever S < S. The increasing subspaces generate a spectral measure E defined for intervals a < x b by E(a,b] = orthogonal projection on Hb Ha. We shall say that E has multiplicity 1 if there exists an element w L2(B,μ) such that {E(σ)w : σ ∈ℬ} spans L2(B,μ). Here denotes the class of Borel subsets of R.

THEOREM 1. Assume that

  1. SHS = {0}.
  2. E has multiplicity 1.

Then μ sits on a coset of R in B.

Mathematical Subject Classification
Primary: 42.50
Received: 6 June 1966
Published: 1 November 1967
Mahendra Ganpatrao Nadkarni