Some nonabsolutely convergent Lebesgue-type integrals

Abstract

As a general tool for integration, the Lebesgue integral is very useful. Its definition is quite general and it has a well-developed theory. However, it does have its limitations. An important limitation is that a function will be Lebesgue integrable only if its absolute value has finite integral, and there exist simple examples of functions that do not satisfy this property, yet intuition suggests they should be integrable. The generalized Riemann integral has helped to solve this problem. Unfortunately, it has a useful theory only for integration over subsets of finite-dimensional Euclidean space. This thesis introduces a new integral, the generalized Lebesgue integral, which can be defined on any sigma-finite measure space, and allows the integration of some functions whose absolute values have infinite integrals. The definition retains some of the flavor of the definition of the Lebesgue integral, and introduces two new concepts: expanding sequences, which are a generalization of monotonic sequences, and semiuniform convergence, which is a weak form of uniform convergence. In addition, to help structure the presentation, an abstract definition of measure-based integrals, called the abstract mu-integral, is introduced.