Multivariable Advanced Calculus

7.5. MEASURES AND OUTER MEASURES 165such thatµ k (K k ) + ε/2 k+1 > µ k (F ∩ (B (0, k) \ D (0, k − 1)))= µ k (F ) = µ (F ∩ (B (0, k) \ D (0, k − 1))) .Since K k is a subset of F ∩ (B (0, k) \ D (0, k − 1)) it follows µ k (K k ) = µ (K k ). Therefore,µ (F ) = lwhenever m is large enough. Therefore, letting µ m (A) ≡ µ (A ∩ B (0, m)) , there existsa compact set, K ⊆ F ∩ B (0, m) such thatµ (K) = µ m (K) > µ m (F ∩ B (0, m)) = µ (F ∩ B (0, m)) > lThis proves the theorem. 7.5 Measures And Outer Measures7.5.1 Measures From Outer MeasuresEarlier an outer measure on P (R) was constructed. This can be used to obtain ameasure defined on R. However, the procedure for doing so is a special case of a generalapproach due to Caratheodory in about 1918.Definition 7.5.1 Let Ω be a nonempty set and let µ : P(Ω) → [0, ∞] be anouter measure. For E ⊆ Ω, E is µ measurable if for all S ⊆ Ω,µ(S) = µ(S \ E) + µ(S ∩ E). (7.7)To help in remembering 7.7, think of a measurable set, E, as a process which dividesa given set into two pieces, the part in E and the part not in E as in 7.7. In the Bible,there are several incidents recorded in which a process of division resulted in more stuffthan was originally present. 2 Measurable sets are exactly those which are incapable ofsuch a miracle. You might think of the measurable sets as the nonmiraculous sets. Theidea is to show that they form a σ algebra on which the outer measure, µ is a measure.First here is a definition and a lemma.2 1 Kings 17, 2 Kings 4, Mathew 14, and Mathew 15 all contain such descriptions. The stuff involvedwas either oil, bread, flour or fish. In mathematics such things have also been done with sets. In thebook by Bruckner Bruckner and Thompson there is an interesting discussion of the Banach Tarskiparadox which says it is possible to divide a ball in R 3 into five disjoint pieces and assemble the piecesto form two disjoint balls of the same size as the first. The details can be found in: The Banach TarskiParadox by Wagon, Cambridge University press. 1985. It is known that all such examples must involvethe axiom of choice.