Multiplicity and the area of an (n− 1) continuous mapping

Ronald Gariepy
1973 Pacific Journal of Mathematics  
For a class of mappings considered by Goffman and Ziemer [Annals of Math. 92 (1970) ] it is shown that the area is given by the integral of a multiplicity function and a convergence theorem is proved. l Introduction. A theory of surface area for mappings beyond the class of continuous mappings was initiated in [2] . This theory includes certain essentially discontinuous mappings for which it seems natural that the area be given by the classical integral formula. Let Q = R n Π {x: 0 < x t < 1
more » ... 1 £ i ^ n}. For each i e {1, , n} and r e 1= {ί: 0 < t < 1} let P^r) = Q Π {x: α>< = r}. A mapping /: Q-»R m , n<^m, is said to be n -1 continuous if, for each i,f\ Pi(r) is continuous for almost every (in the sense of 1-dimensional Lebesgue measure) re I. A sequence {f ό } of mappings from Q into R m is said to converge n -1 to / if, for each i, f s \ Pi(r) converges uniformly to /1 Pi(r) for almost every r el. The area of an n -1 continuous mapping /: Q->R m is defined as A(f) =inflimα(/ y )
doi:10.2140/pjm.1973.44.509 fatcat:bqlx5dibhrbwfkvmqgl3jgfvei