  A multiple operator integral (MOI) is an indispensable tool in several
branches of noncommutative analysis. However, there are substantial technical
issues with the existing literature on the "separation of variables" approach
to defining MOIs, especially when the underlying Hilbert spaces are not
separable. In this paper, we provide a detailed development of this approach in
a very general setting that resolves existing technical issues. Along the way,
we characterize several kinds of "weak" operator valued integrals in terms of
easily checkable conditions and prove a useful Minkowski-type integral
inequality for maps with values in a semifinite von Neumann algebra.
