The variable exponent Sobolev capacity and quasi-fine properties of Sobolev functions in the case p−=1

In this article we extend the known results concerning the subadditivity of capacity and the Lebesgue points of functions of the variable exponent Sobolev spaces to cover also the case p−=1. We show that the variable exponent Sobolev capacity is subadditive for variable exponents satisfying 1⩽p<∞. Furthermore, we show that if the exponent is log-Hölder continuous, then the functions of the variable exponent Sobolev spaces have Lebesgue points quasieverywhere and they have quasicontinuous representatives also in the case p−=1. To gain these results we develop methods that are not reliant on reflexivity or maximal function arguments.