Let A be a Cartan subalgebra of a von Neumann algebra M. This means A is a masa in M, the set of unitaries \(u\in M\) satisfying \(u^{-1}Au=A\) generates M, and there is a faithful normal expectation from M onto A. The simplest example has \(M=M_ n({\mathbb{C}})\) with A its subalgebra of diagonal matrices. In their papers [Trans. Amer. Math. Soc. 234, 289-324 (1977; Zbl 0369.22009 and 369.22010)], \textit{J. Feldman} and \textit{C. Moore} developed a matricial point of view for the general situation. There exists a standard Borel equivalence relation R on a set X so that members of M can be identified with matrices indexed by R acting by (possibly twisted) matrix multiplication on a suitable \(L^ 2(R,v)\) space; members of A correspond to matrices supported in the diagonal \(\Delta\) in R. Suppose now that \({\mathcal G}\) is a \(\sigma\)-weakly closed linear subspace of M that is a bimodule over A. Theorem 2.5 of the present paper states that there is a Borel subset B of R such that \({\mathcal G}\) consists of all those operators whose matrices are supported on B. While this is obvious in the \(M_ n({\mathbb{C}})\) example, the general argument is quite delicate, not only for measure-theoretic reasons, but primarily because simple changes in a matrix stop it from representing a bounded operator. Armed with their bimodule spectral theorem, the authors then proceed to relate properties of \({\mathcal G}\) to those of the support relation B. In particular, \({\mathcal G}\) is a maximal triangular (\({\mathcal G}\cap {\mathcal G}\) \(*=A)\sigma\)-weakly closed subalgebra of M iff B is (essentially) a partial order that totally orders each R equivalence class. This repesents a fairly complete generalization of Theorem 3.21 of \textit{R. Kadison} and \textit{I. Singer}'s pioneering paper [Amer. J. Math. 82, 227- 259 (1960; Zbl 0096.317)]. In the \(M_ n({\mathbb{C}})\) example, \(R=\{1,2,...,n\}\times \{1,2,...,n\}\), and a total order \(B\subseteq R\) allows us to rearrange our basis vectors so that matrices of members of \({\mathcal G}\) become upper triangular.