Vol. 23, No. 2, 1967

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
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