will be a σ finite measure space as usual.
is strongly measurable as a map into Y.
x must be strongly measurable.
∫ Ψ (x) = Ω A (s)xdμ.
dμ and the definition is well defined.
∗ ||y|| = ys∗u∈pD′|y (y)|.
x is strongly, hence weakly measurable.
∫ Ω||A (s)x||dμ < C ||x||.
is compact and self adjoint. Recall the Kuratowski measurable selection theorem, Theorem 9.1.11 on Page 598 listed here for convenience.
ψ(f (ω ),ω) = sxu∈pE ψ(x,ω).