Integral Representation of Linear Functionals

Abstract

This chapter is devoted to questions relevant to both measure theory and functional analysis. In Sect. 12.1, we develop a general scheme allowing one to obtain integral representations for order continuous functionals on functional spaces, in particular, in the spaces $\mathcal{L}^{p}$ for finite p. In Sect. 12.2, we prove that a positive functional on the space of continuous functions defined on a locally compact space has an integral representation. In Sect. 12.3, we describe the general form of continuous linear functionals in the spaces of functions continuous on a compact spaces and also in the spaces $\mathcal{L}^{p}$ for 1⩽p<∞. As a consequence, we prove that the Borel charges on a multi-dimensional torus are determined by their Fourier coefficients. We consider various applications of these results to harmonic analysis.