Lebesgue’s theorem of differentiation in Fréchet lattices

Abstract

Lebesgue’s differentiation theorem (LDT) states that every monotonic real function is differentiable a.e. We investigate the validity of this theorem for functions with values in topological vector lattices. It is shown that a Fréchet lattice satisfies (LDT) iff it is isomorphic to a generalized echelon space, a Banach lattice satisfies (LDT) iff it is isomorphic to some l 1 ( Γ ) {l^1}\left ( \Gamma \right ) .