The talk gives some survey about recent applications of finitely additive measures to Lebesgue integrable functions. After a short introduction to such measures and related integrals, purely finitely additive measures are of particular interest. Special examples are given and, as a first application, an integral representation for the precise representative of Lebesgue integrable functions is provided. Then, based on a general approach to traces, a new version of the Gauss-Green formula is introduced, where neither a pointwise trace nor a pointwise normal is needed on the boundary. This allows e.g. the treatment of inner boundaries and of concentrations on the boundary. A second boundary integral is used to handle singularities that hadnot been accessible before. Finally, weak versions of differentiability for Lebesgue integrable functions are discussed, a mean value formula for a class of Sobolev functions is given, and a new approach to the generalized derivatives in the sense of Clarke is provided.
