1 On the Lebesgue integral and the Lebesgue measure revisited: mathematical applications in some sectors of Chern-Simons theory and Yang-Mills gauge theory and mathematical connections with some sectors of String Theory, Supersymmetry Breaking and Number Theory Michele Nardelli 2 , 1 1 Dipartimento di Scienze della Terra Università degli Studi di Napoli Federico II, Largo S. Marcellino, 10 80138 Napoli, Italy 2 Dipartimento di Matematica ed Applicazioni “R. Caccioppoli” – Università degli Studi di Napoli “Federico II” – Polo delle Scienze e delle Tecnologie Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy Abstract In this paper, in the Section 1, we have described some equations and theorems concerning the Lebesgue integral and the Lebesgue measure. In the Section 2, we have described the possible mathematical applications, of Lebesgue integration, in some equations concerning various sectors of Chern-Simons theory and Yang-Mills gauge theory, precisely the two dimensional quantum Yang-Mills theory. In conclusion, in the Section 3, we have described also the possible mathematical connections with some sectors of String Theory and Number Theory, principally with some equations concerning the Ramanujan‟s modular equations that are related to the physical vibrations of the bosonic strings and of the superstrings, some Ramanujan‟s identities concerning π and the zeta strings. 1. On the Lebesgue integral and the Lebesgue measure [1] [2] [3] [4] In this paper we use Lebesgue measure to define the of functions , where is the Euclidean space. If is an unsigned simple function, the integral is defined by the formula