Pointwise selection theorems for metric space valued bivariate functions

We introduce a pseudometric TV on the set MX of all functions mapping a rectangle X on the plane R2 into a metric space M, called the total joint variation. We prove that if two sequences {fj} and {gj} of functions from MX are such that {fj} is pointwise precompact on X, {gj} is pointwise convergent on X with the limit g∈MX, and the limit superior of TV(fj,gj) as j→∞ is finite, then a subsequence of {fj} converges pointwise on X to a function f∈MX such that TV(f,g) is finite. One more pointwise selection theorem is given in terms of total ε-variations (ε>0), which are approximations of the total variation as ε→0.