On the Aumann–Shapley value

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Let L be a D-lattice, i.e. a lattice ordered effect algebra, and let BV be the Banach space of all real-valued functions of bounded variation on L (vanishing at 0) endowed with the variation norm. We prove the existence of a continuous Aumann–Shapley value φ on \(\mathfrak{bv}^\prime\)NA, the subspace of BV spanned by all functions of the form \(f\circ\mu\), where \(\mu: L \to [0,1]\) is a non-atomic σ-additive modular measure and \(f: [0,1] \to {\mathbb{R}}\) is of bounded variation and continuous at 0 and at 1.