We show how to integrate operator valued functions with respect to a spectral or orthogonally scattered measure. Such measures typically have a variation which has either the value zero or infinity on any set and cannot therefore be treated by the approaches of Bartle or Dobrakov. Bilinear integrals of this type arise from trace class operators between Banach function spaces and in the connection between stationary-state scattering theory and time-dependent scattering theory in Hilbert space.
Brian Jefferies. "Bilinear integration and applications to operator and scattering theory." Rocky Mountain J. Math. 46 (1) 189 - 225, 2016. https://doi.org/10.1216/RMJ-2016-46-1-189