δ-ring representation of lattice homomorphisms on L p -Spaces

We study bounded lattice homomorphisms from L p -spaces over complete σ-finite measure spaces into Banach lattices. For 1 ≤ p < ∞, we work on the δ-ring of finitemeasure sets and on simple functions with finite-measure support. We show that such operators are determined by their values on indicator functions of finite-measure sets, which yields a component-valued set function satisfying a local Boolean property. The operator is recovered as the unique bounded extension of the associated integral on simple functions. We obtain an exact operator-norm formula in terms of a natural size functional of the induced set function and prove a converse construction. In the AL-space case we derive a Radon–Nikodým description and an explicit norm identity. Endpoint variants for p = 1 on finite measure spaces and for p = ∞ under a σ-order continuity hypothesis are also included.