In this paper, firstly, we obtain some new results about bornological convergence in locally convex cones (which was studied in [1]) and then we introduce the concept of bornological completion for locally convex cones. Also, we prove that the completion of a bornological locally convex cone is bornological. We illustrate the main result by an example.

We characterize bounded sets and bounded operators in locally convex cones. Also, we study some relations between barreledness and boundedness in locally convex cones.

We consider the locally convex product cone topologies and prove that the product topologyof weakly cone-complete locally convex cones is weakly cone-complete. In particular, we deduce that a product cone topology is barreled whenever its components are weaklycone-complete and carry the countable neighborhood bases.

In this paper, we define the almost uniform convergence and the almost everywhere convergence for cone-valued functions with respect to an operator valued measure. We prove the Egoroff theorem forP-valued functions and operator valued measure q: ®L (P, Q), where R is a s-ring of subsets of X¹q, (P, V) is a quasi-full locally convex cone and (Q, W) is a locally convex complete lattice cone.