Abstract: | Let (Ω,
, μ) be a measure space,
a separable Banach space, and
* the space of all bounded conjugate linear functionals on
. Let f be a weak* summable positive B(
*)-valued function defined on Ω. The existence of a separable Hilbert space
, a weakly measurable B(
)-valued function Q satisfying the relation Q*(ω)Q(ω) = f(ω) is proved. This result is used to define the Hilbert space L2,f of square integrable operator-valued functions with respect to f. It is shown that for B+(
*)-valued measures, the concepts of weak*, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained. |