# 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.

## Keywords

Measurable Function Topological Space Measure Space Countable Union Monotone Convergence Theorem## Preview

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