Lebesgue Integration

Abstract

In Chapter II we constructed Lebesgue’s measure on ${\mathbb{R}^N}$. The result of our efforts was a proof that there is a class ${\overline B _{{\mathbb{R}^N}}}$ of subsets of ${\mathbb{R}^N}$ and a map $\Gamma \in {{\bar{B}}_{{{{\mathbb{R}}^{N}}}}} \mapsto \left| \Gamma \right| \in \left[ {0,\infty } \right]$ such that: ${{\bar{B}}_{{{{\mathbb{R}}^{N}}}}}$ contains all open sets; ${{\bar{B}}_{{{{\mathbb{R}}^{N}}}}}$ is closed under both complementation and countable unions; $\left| I \right| = vol\left( I \right)$ for all rectangles I; and $\left| {{ \cup _1}\infty {\Gamma _n}} \right| = \sum\limits_1^\infty {\left| {{\Gamma _n}} \right|}$ whenever $\left\{ {{\Gamma _n}} \right\}_1^\infty$ is a sequence of mutually disjoint elements of ${{\bar{B}}_{{{{\mathbb{R}}^{N}}}}}$. What we are going to do in this section is discuss a few of the general properties which are possessed by such structures.