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Lebesgue Integration

  • Daniel W. Stroock

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Daniel W. Stroock 1994

Authors and Affiliations

  • Daniel W. Stroock
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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