Quantum-Logics-Valued Measure Convergence Theorem

Abstract

In this paper, the following quantum-logic valued measure convergence theorem is proved: Let (L ₁, 0, 1) be a Boolean algebra, (L ₂, ⊥, ⊕, 0, 1) be a quantum logic and {μ _n : n ∈ N} be a sequence of s-bounded (L ₂, ⊥, ⊕, 0, 1)-valued measures which are defined on (L ₁, 0, 1). If for each a ∈ (L ₁, 0, 1), {μ _n (a)} _{n ∈ N} is an order topology $\tau_{0}^{L_2}$ Cauchy sequence, when {v(a)} convergent to 0, {μ _n (a)} is order topology $\tau_{0}^{L_2}$ convergent to 0 for each n ∈ N, where v is a nonnegative finite additive measure which is defined on (L ₁, 0, 1), then when {v(a)} convergent to 0, {μ _n (a)} are order topology $\tau_{0}^{L_2}$ convergent to 0 uniformly with respect to n ∈ N.