Measure Theory and Integration
This text approaches integration via measure theory as opposed to measure theory via integration, an approach which makes it easier to grasp the subject. Apart from its central importance to pure mathematics, the material is also relevant to applied mathematics and probability, with proof of the mathematics set out clearly and in considerable detail. Numerous worked examples necessary for teaching and learning at undergraduate level constitute a strong feature of the book, and after studying statements of results of the theorems, students should be able to attempt the 300 problem exercises which test comprehension and for which detailed solutions are provided.
Approaches integration via measure theory, as opposed to measure theory via integration, making it easier to understand the subjectIncludes numerous worked examples necessary for teaching and learning at undergraduate levelDetailed solutions are provided for the 300 problem exercises which test comprehension of the theorems provided
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Measure on the Real Line
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absolutely continuous apply Borel sets bounded Chapter choose Clearly closed complete condition consider construction contains continuous function convergence convex Corollary corresponding countable cover decomposition defined Definition denote disjoint equality Example Exercise exists extended f and g f dx f is measurable finite fn(x follows function f given gives the result hence holds implies inequality integrable function intervals length Let f lim inf lim sup limit mean measurable function measurable set measure space monotone increasing non-negative notation o-algebra o-finite measure obtain obvious open intervals open set outer measure partition points positive Proof properties prove respect result follows Riemann integrable sequence Show side signed measure Similarly Solution subsets suppose Theorem 11 Theorem 9 uniformly union unique values write