4 - Measure and Integration, Function Spaces

Abstract

The most essential groundwork for the notion of “abstract measure” and the theory of integration was laid by E. Borel and H. Lebesgue in the early 20th century. Abstract measures are founded upon the notion of measurable sets (sets equipped with a σ-algebra T) and play a key role in probability theory. Radon measures (which were formulated by F. Riesz and J. Radon between 1909 and 1913) and their properties are studied. These measures are based on the theory of duality in locally convex spaces. Abstract Borel measures are equivalent to positive Radon measures on locally compact topological spaces. Each type of measure has its own advantages and challenges: abstract measures may be defined in a more general context than Radon measures, and in particular calculating probabilities is extremely difficult with the latter. The theory of Radon measures is straightforward on locally compact topological spaces, but becomes more intricate when working with general topological spaces. On the other hand, the images of abstract measures exhibit pathological behavior in ways that the images of Radon measures don’t; for more details. Radon measures are distributions of order 0.