Abstract
In this chapter we provide a variety of extension and representation results for linear functionals defined on spaces of random variables. The basic extension result states that every linear functional defined on a proper subspace of random variables can be extended to the entire space of random variables preserving linearity. The main representation result is a version of the classical Riesz representation, which states that every linear functional defined on a space of random variables can be represented in terms of an expectation. We pay special attention to linear functionals that are strictly positive because the corresponding extension and representation results play a fundamental role in the study of financial markets.
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