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 ℬ denotes the class of Borel subsets of
R.
