A Generalization of the Integral Mean Value Theorem from the Perspective of Two-Dimensional Lebesgue Measure
Year: 2025
Type: preprint
Abstract: This paper proposes and investigates the concept of "integral conservation" within the framework of twodimensional Lebesgue measure, aiming to understand the essence of the integral mean value theorem from a structural perspective. By identifying each one-dimensional interval a,b with a point (a,b) in the upper half-plane, we construct a mapping of the function's integrals over all possible intervals and analyze its geometric and measure-theoretic properties. We prove that for essentially bounded Lebesgue integrable functions, the average value over any interval must lie between the essential infimum and supremum of the function, thereby ensuring the existence of a "representative value" that realizes integral conservation. Furthermore, when the function is continuous or satisfies the intermediate value property, this representative value can be achieved as the function value at some mean-value point. Leveraging the Lebesgue differentiation theorem, we also establish a connection between integral conservation and local averaging behavior, providing a weak reconstruction of the differential mean value theorem. This work demonstrates that the integral mean value theorem is not merely a pointwise identity, but rather a manifestation of a deeper structural invariance-the "integral trajectory" of the function over families of intervals remains conserved, and the mean-value point serves as its geometric representative. This framework offers a new measure-theoretic perspective for studying the global integral behavior of functions.
Author: Hongtao Ling
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4414871010