Liapunov theorem for modular functions (Q1907540)

Generalizing the classical convexity theorem of Liapunov, the author proves as main theorem: If \(\mu: L\to \mathbb{R}^n\) is a nonatomic modular function on a complemented lattice satisfying the interpolation property (a condition weaker than \(\sigma\)-completeness), then \(\mu([0, a])\) is convex for any \(a\in L\). Hereby, complemented lattices of particular interest are orthomodular lattices.