Lebesgue Integral の理論

最近フーリエー級数を勉強する時にLebesgue積分の事を呼んでいます。
I found this somewhat confusing, as I had always believed that an integral, in two dimensions the area under the curve, had a single definition as explored by Newton and Leibniz. In addition, I had believed that the standard way to derive the integral was in using Riemann sums, rectangular pieces which approximate the curve over an interval, and whose areas can be summed. However, it seems that this "Riemann sums" approximation method is based on the "Riemann integral". Some functions cannot be integrated using this definition, according to what I have read. Thus, a different definition, formulated by Henri Lebesgue in 1902, is better suited to integration of these nonstandard functions.
This begs the question, how is the Lebesgue integral actually defined? The integral is said to be valid on functions which are "measurable", meaning that for all a0. I am not sure whether the "total volume" here refers to the individual intervals or to the product. A better way to derive this seems to be to consider a single point. The measure of a single point defaults to zero. Because measure is preserved under countable addititivity, the measure of a set containing a countable number of points is the sum of the measures of those points and is thus zero.
Another problem is with the definition of sigma algebra. It seems that Lebesgue integrable functions are defined on subsets of a set which are closed on a countable numer of unions, intersections, and complements. I am not sure why these conditions are necessary. I assume it has something to do with the properties like the additivity of the measure, since, if the subsets were not closed under union, the sum of the measures of two points might not equal the measure of the union.
Many examples in theorems also use the abbreviations "sup" and "inf" with which I am unfamiliar. For finite sets, sup (supremum) and inf (infimum) are the same as max and min. sup(2,5,10)=2 and inf(2,5,10) = 2. However, for infinite sets, they refer to the least upper bound and the greatest lower bound. e.g. for real numbers 0