Last years many results are obtained in non commutative (quantum) probability theory on von Neumann algebras. Usually their proofs exploit complicated techniques from von Neumann algebra theory. One of the useful methods for obtaining similar results in JW-algebras (Jordan algebras of self-adjoint operators) is the reduction to the case of von Neumann algebras. Therefore it is natural to study the connection between various notions in JW-algebras and their enveloping von Neumann algebras. In the present paper the author considers such connection for the notions of convergence almost everywhere and almost completely. As a corollary a strong law of large number is obtained in JW-algebras.