Introdução

It is well known that the Riemann integral is not adequate for advanced mathematics, since there are many functions that are not Riemann integra-
ble, and since the integral does not posses sufficiently strong convergence theorems. To get rid of these deficiencies, Lebesgue integral came into mathematics in the turn of the present century and from there on has become the official integral in mathematical research. Many good books have benn written on the subject. Most of then have the problem of being to general for an introductory course, in the sense that present general Lebesgue measure in very abstract spaces. We believe there is still room for a presentation of Lebesgue measure in making strongly use of the fact we are dealing with . So many proofs could be presented in a shorter and more geometric form. Moreover most of the books lack the discussion of the classical theorems of vector analysis $\mathbb{R}$ $\mathop{\mbox{\ensuremath{\int\!\int } }}\limits_{\hskip -2pt U}$