32.4. MEASURES FROM OUTER MEASURES 639

To help in remembering 32.4, think of a measurable set E, as a process which divides agiven set into two pieces, the part in E and the part not in E as in 32.4. In the Bible, thereare several incidents recorded in which a process of division resulted in more stuff thanwas originally present.1 We don’t want this. Measurable sets are exactly those which areincapable of such a miracle. You might think of the measurable sets as the non-miraculoussets. The idea is to show that they form a σ algebra on which the outer measure µ is ameasure.

First here is a definition and a lemma.

Definition 32.4.2 (µ⌊S)(A) ≡ µ(S∩A) for all A ⊆ Ω. Thus µ⌊S is the name of anew outer measure, called µ restricted to S.

The next lemma indicates that the property of measurability is not lost by consideringthis restricted measure.

Lemma 32.4.3 If A is µ measurable, then A is µ⌊S measurable.

Proof: Suppose A is µ measurable. It is desired to to show that for all T ⊆ Ω,

(µ⌊S)(T ) = (µ⌊S)(T ∩A)+(µ⌊S)(T \A).

Thus it is desired to show

µ(S∩T ) = µ(T ∩A∩S)+µ(T ∩S∩AC). (32.5)

But 32.5 holds because A is µ measurable. Apply Definition 32.4.1 to S∩T instead of S.

If A is µ⌊S measurable, it does not follow that A is µ measurable. Indeed, if you believein the existence of non measurable sets, you could let A = S for such a µ non measurableset and verify that S is µ⌊S measurable.

The next theorem is the main result on outer measures which shows that starting withan outer measure you can obtain a measure.

Theorem 32.4.4 Let Ω be a set and let µ be an outer measure on P (Ω). Thecollection of µ measurable sets S , forms a σ algebra and

If Fi ∈ S, Fi ∩Fj = /0, then µ(∪∞i=1Fi) =

∞

∑i=1

µ(Fi). (32.6)

If · · ·Fn ⊆ Fn+1 ⊆ ·· · , then if F = ∪∞n=1Fn and Fn ∈ S , it follows that

µ(F) = limn→∞

µ(Fn). (32.7)

If · · ·Fn ⊇ Fn+1 ⊇ ·· · , and if F = ∩∞n=1Fn for Fn ∈ S then if µ(F1)< ∞,

µ(F) = limn→∞

µ(Fn). (32.8)

This measure space is also complete which means that if µ (F) = 0 for some F ∈ S thenif G ⊆ F, it follows G ∈ S also.

11 Kings 17, 2 Kings 4, Mathew 14, and Mathew 15 all contain such descriptions. The stuff involved waseither oil, bread, flour or fish. In mathematics such things have also been done with sets. In the book by BrucknerBruckner and Thompson there is an interesting discussion of the Banach Tarski paradox which says it is possibleto divide a ball in R3 into five disjoint pieces and assemble the pieces to form two disjoint balls of the samevolume as the first. The details can be found in: The Banach Tarski Paradox by Wagon, Cambridge Universitypress. 1985. It is known that all such examples must involve the axiom of choice.