Nullfunctions and the Lebesgue Spaces L_p

Abstract

The integral with respect to a submodular set function allows to define norms on spaces of measurable functions. An obstacle in doing so is that (like in the σ-additive theory) there are functions, not identically zero, having norm nought. Those are the nullfunctions, we start with. Then the Lebesgue space L ₁(µ) of a submodular µ is shown to be a normed linear space and to be a Banach space if µ is continuous from below. These results translate to L _∞(µ) via the identity L _∞(µ) = L ₁(sign µ). The spaces L _p , 1 < p < ∞, are defined but not treated in detail. At the end of the chapter we show that assigning the quantile function to a function is a continuous even contracting operator χ. Recall that this assignment was the first step in Choquet’s and our approach to integration theory which now turns out to have good topological and metric properties. Furthermore χ is piecewise linear, namely on cones of comonotonic functions.