Vector measure

Summary

A vector measure is an additive set function whose values lie in a Banach space instead of the real or complex numbers. The basic notion is finite additivity; countable additivity is an extra property, with convergence of series in norm. Vector measures arise from spectral theory, where a self-adjoint operator gives a projection-valued measure, and from integration of vector-valued functions. Their theory, including the Bartle–Dunford–Schwartz integral and Lyapunov's convexity theorem, studies how far scalar measure theory extends to infinite dimensions.