Measure on a Lattice of Infinite Tuples of Real Numbers

We consider the analogue of measure on a ring of sets, by defining a lattice-measure on a lattice-ring of elements of a lattice of infinite tuples of real numbers. We obtain various results about the convergence of sequences of tuples in the lattice with respect to the lattice-measure and use these results to show that the limit of a convergent sequence in a lattice sigma-ring is also in the lattice sigma-ring under certain conditions.