The problems of characterizing the class of functions that are Riemann integrable and of discussing discontinuous functions, in particular of understanding for which functions the fundamental theorem of calculus is valid, as well as the need of integrating new functions, led to a new definition of integral due to Henri Lebesgue (1875–1941). Though the main ideas of Lebesgue's integration theory go back to Henri Lebesgue (1875–1941) and Giuseppe Vitali (1875–1932) at the beginning of the 1900's, applications as well as generalizations and extensions followed each other during the past century giving measure and integration theory a fundamental role in mathematical analysis. Here we follow the approach of first introducing Lebesgue's measure and accordingly Lebesgue's integral. In Section 2.1.1 we collect the main results of the theory without proofs,1 and in the following sections we develop its basic features

## Keywords

Measurable Subset Lebesgue Point Green Formula Polyhedral Surface Integral Calculus## Preview

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