The original purpose of this paper was to establish that certain non-parametric variational integrals may be considered as measures on the domain of definition of the admissible functions. This was accomplished by exhibiting an explicit formula for the functional involved, the formula containing within itself, in the case of the surface area integral, a proof both of Tonelli's celebrated theorem on Lebesgue area and of a less well known but deeper result of Verchenko. E . This set function is in some ways analogous to the total variation of ;

Abstract. We shall show that the space of all approximately continuous functions with the topology of pointwise convergence is not homeomorphic to its category analogue.2000 AMS Classification: 54C35, 54C30, 26A15.

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