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

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 \( \mathbb{R} \)?, where by a Lebesgue class (corresponding to some \( x \in \mathbb{R} \)) 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.