Lebesgue Density Theorem

Calculus is a branch of mathematics in which the differential and the integral of thefunctions and the related concepts and applications are studied. There were already manyintuitive and useful integrals in19th century, e.g. Riemmann integral, Riemann-Stieltjesintegral etc. While there are big problems for these integrals when the exchange of theorder of the integrals and limits are concerned. To release the constructions, the Frenchmathematician Lebesgue invented the so-called Lebesgue integral in1902. The Lebesgue-Stieltjes integral was then established, and the theory of measure and integral was thus inshaping.Measure theory is a further abstraction and development of Lebesgue measure andLebesgue integral theory, and it plays an important role in modern analysis. In1910,Lebesgue established the so-called Lebesgue differential theorem. And the Lebesgue den-sity theorem, which is one of the main objects studied in this thesis, can be regarded as aspecial case of the differential theorem.In the introductory chapter we mainly introduce the background and current situationof the research, and then review some previous work and the conclusion. In the secondchapter, we provide a simple proof of the Lebesgue density theorem. In the last chapter weconsider several related problems of the Lebesgue density theorem, including a converse ofLebesgue density theorem; exceptional points for Lebesgue density theorem; analogues ofthe Lebesgue density theorem for fractal sets in reals and integers; and Lebesgue densitytheorem in infinite-dimensional spaces. We point out that in an infinite dimensional space,the counterpart of Lebesgue density theorem has not yet been conclusive.