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The purpose of this note is to answer a question A. E. Nussbaum formulated in1964 about the possible equivalence between weak measurability of a family ofdensely defined, closed operators T(t), t real, in a separable complex Hilbertspace H on one hand, and the notion of measurability of the 2 \times 2operator-valued matrix of projections onto the graph Gamma(T(t)) of T(t) on theother, in the negative.  Our results demonstrate an interesting distinction between the directintegral over the family of operators T(t) with respect to Lebesgue measure andthe naturally maximally defined operator associated with pointwise applicationof 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 ofprojections onto the graph Gamma(T(t)) of T(t).

On a Question of A. E. Nussbaum on Measurability of Families of Closed Linear Operators in a Hilbert Space