Integrable Functions Versus a Generalization of Lebesgue Points in Locally Compact Groups

The author is thankful to the referee for his valuable comments and suggestions that led to an improvement of the paper. He also owes to Prof. M. N. Mukherjee of the Deptt. of Pure Mathematics, Calcutta University, for the present linguistically improved version.Here in this paper we intend to deal with two questions: How large is a “Lebesgue Class” in the topology of Lebesgue integrable functions, and also what can be said regarding the topological size of a “Lebesgue set” in R?, where by a Lebesgue class (corresponding to some x in R) is meant the collection of all Lebesgue integrable functions for each of which the point x acts as a common Lebesgue point, and, by a Lebesgue set (corresponding to some Lebesgue integrable function f ) we mean the collection of all ebesgue points of f. However, we answer these two questions in a more general setting where in place of Lebesgue integration we use abstract integration in locally compact Hausdorff topological groups