  The purpose of this note is to answer a question A. E. Nussbaum formulated in
1964 about the possible equivalence between weak measurability of a family of
densely defined, closed operators T(t), t real, in a separable complex Hilbert
space H on one hand, and the notion of measurability of the 2 \times 2
operator-valued matrix of projections onto the graph Gamma(T(t)) of T(t) on the
other, in the negative.
  Our results demonstrate an interesting distinction between the direct
integral over the family of operators T(t) with respect to Lebesgue measure and
the naturally maximally defined operator associated with pointwise application
of T(t) in the vector-valued Hilbert space L^2(\mathbb R; dt; H).
  We also provide explicit criteria for the measurability of the matrix of
projections onto the graph Gamma(T(t)) of T(t).
