A Note on the Lower Semi-continuity of Double Integrals in the Parametric Form

where x stands for the three coordinate functions, ^(w , u)} ff (#\ u), x{u, u), and X stands for their three Jacobians. Various theorems of the following general type have been proved by McShane, Caccioppoli, Cimmino, and Radó: If (i) the triples of functions xn{u , u), i = 1, 2; n = 0, 1, 2, • • • , satisfy certain conditions; (ii) the surfaces defined by x = xn(u , u), i = l,2;n = l,2, • • • , converge in some prescribed sense to the surface defined by x^x^u, u), i = l, 2; (iii) f(x, X) satisfies certain conditions; then