The Borel hierarchy classifies subsets of the reals by their topological complexity. Another approach is to classify them by size.

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1 Lecture 7: Measure ad Category The Borel hierarchy classifies subsets of the reals by their topological complexity. Aother approach is to classify them by size. Filters ad Ideals The most commo measure of size is, of course, cardiality. I the presece of ucoutable sets (like i a perfect Polish space), the usual divisio is betwee coutable ad ucoutable sets. The smalless of the coutable sets is reflected, i particular, by two properties: A subset of a coutable set is coutable, ad coutable uios of coutable set are coutable. These characteristics are shared with other otios of smalless, two of which we will ecouter i this lecture. Defiitio 7.1: A o-empty family I P(X ) of subsets of a give set X is a ideal if (I1) A I ad B A implies B I, (II2) A, B I implies A B I. If we have closure eve uder coutable uios, we speak simply of a σ-ideal. For example, while the coutable sets i form a σ-ideal, the fiite subsets oly form a ideal. Aother example of ideals are the so-called pricipal ideals. These are ideals of the form Z = {A: A Z} for a fixed Z X. The dual otio to a ideal is that of a filter. It reflects that the sets i a filter share some largeess property. Defiitio 7.2: A o-empty family F P(X ) of subsets of a give set X is a σ-filter if (F1) A F ad B A implies B F, (F2) A, B F for all implies A B F. Agai, closure uder coutable itersectio yields σ-filters. 7 1

2 If I is a (σ-) ideal, the F = { A: A I} is a (σ-) filter. Hece the co-fiite subsets of form a filter, ad the co-coutable subsets form a σ-filter. Note that the complemet of a (σ-) ideal (i P(X )) is ot ecessarily a (σ-) filter. This is true, however, for a special class of ideals/filters. Defiitio 7.3: A o-empty family I P(X ) is a prime ideal if it is a ideal for which for every A X, either A I or A I. A ultrafilter is a filter whose complemet i P(X ) is a prime ideal. I light of the small-/largeess motivatio, prime ideals ad ultrafilters provide a complete separatio of X : Each set is either small or large. Measures Coarsely speakig, a measure assigs a size to a set i a way that reflects our basic geometric ituitio about sizes: The size of the uio of disjoit objects is the sum of their sizes. The questio whether this ca be doe i a cosistet way for all subsets of a give space is of fudametal importace ad has motivated may questios i set theory. The formally, a measure µ o X is a [0, ]-valued fuctio defied o subsets of X that satisfies (M1) µ()=0, (M2) µ( A )= µ(a ), wheever the A are pairwise disjoit. The questio is, of course, which subsets of X ca be assiged a measure. The coditio (M2) suggests that this family is closed uder coutable uios. Furthermore, if A X, the the equatio µ(x )=µ(a)+µ( A) suggests that A should be measurable, too. I other words, the sets who are assiged a measure form a σ-algebra. Defiitio 7.4: A measurable space is a pair (X, S), where X is a set ad S is a σ-algebra o X. A measure o a measurable space (X, S) is a fuctio µ : S [0, ] that satis fies (M1) ad (M2) for ay pairwise disjoit family {A } i S. If µ is a measure o (X, S), the the triple (X, S, µ) is called a measure space. 7 2

3 If we wat the measure µ to reflect also some other basic ituitio about geometric sizes, this ofte puts restrictios o the σ-algebra of measurable sets. For example, i the measure of a iterval should be its legth. We will see later (whe we discuss the Axiom of Choice) that it is impossible to assig every subset of a measure, so that (M1) ad (M2) are satisfied, ad the measure of a iterval is its legth. To have some cotrol over what the σ-algebra of measurable sets should be, oe ca costruct a measure more carefully, start with a measure o basic objects such as itervals or balls, ad the exted it to larger classes of sets by approximatio. A essetial compoet i this extesio process is the cocept of a outer measure. Defiitio 7.5: A outer measure o a set X is a fuctio µ : P(X ) [0, ] such that (O1) µ ()=0, (O2) A B implies µ (A) µ (B), (O3) µ ( A ) µ (A ), for ay coutable family {A } is X. A outer measure hece weakes the coditios of additivity (M2) to subadditivity (O3). This makes it possible to have o-trivial outer measures that are defied o all subsets of X. The usefuless of outer measures lies i the fact that they ca always be restricted to subset of P(X ) o which they behave as measures. Defiitio 7.6: Let µ be a outer measure o X. A set A X is µ -measurable if µ (B)=µ (B A)+µ (B \ A) for all B X. This defiitio is a rather obscure. It is justified rather by its cosequeces tha its ituitive appeal. Regardig the latter, suffice it to say here that outer measures may be rather far from beig eve fiitely additive. The defiitio sigles out those sets that split all other sets correctly, with regard to measure. Propositio 7.7: The class of µ -measurable sets forms a σ-algebra M, ad the restrictio of µ to M is a measure. For a proof see for istace Halmos [1950]. 7 3

4 The size of the σ-algebra of measurable sets depeds, of course, o the outer measure µ. If µ is behavig rather pathetically, we caot expect M to cotai may sets. Lebesgue measure A stadard way to obtai ice outer measures is to start with a well-behaved fuctio defied o a certai class of sets, ad the approximate. The paradigm for this approach is the costructio of Lebesgue measure o. Defiitio 7.8: The Lebesgue outer measure λ of a set A is defied as λ (A)=if b a : A (a, b ). Oe ca show that this ideed defies a outer measure. We call the λ - measurable sets Lebesgue measurable. Oe ca verify that every ope iterval is Lebesgue measurable. It follows from Propositio 7.7 that every Borel set is Lebesgue measurable. The costructio of Lebesgue measure ca be geeralized ad exteded to other metric spaces, for example through the cocept of Hausdorff measures. All these measures are Borel measures, i the sese that the Borel measures are measurable. However, there measurable sets that are ot Borel sets. The reaso for this lies i the presece of ullsets, which are measure theoretically easy (sice they do ot cotribute ay measure at all), but ca be topologically quite complicated. Nullsets Let µ be a outer measure o X. If µ (A)=0, the A is called a µ -ullset. Propositio 7.9: Ay µ -ullset is µ -measurable. Proof. Suppose µ (A) =0. Let B X. The, sice µ is subadditive ad mootoe, µ (B) µ (B A)+µ (B A)=µ (B A) µ (B), ad therefore µ (B)=µ (B A)+µ (B A). 7 4

5 The ext result cofirms the ituitio that ullsets are a otio of smalless. Propositio 7.10: The µ -ullsets form a σ-ideal. Proof. (I1) follows directly from mootoicity (O2). Coutable additivity follows immediately from subadditivity (O3). I case of Lebesgue measure, we ca use Propositio 7.9 to further describe the Lebesgue measurable subsets of. Propositio 7.11: A set A is Lebesgue measurable if ad oly if it is the differece of a Π 0 2 set ad a ullset Proof. We first assume λ (A) <. Let G be a ope set such that G A ad λ (G ) λ (A)+1/. The existece of such a G follows from the defiitio of λ, ad the fact that every ope set is the disjoit uio of ope itervals. The G = G is Π 0 2, A G, ad for all, λ (A) λ (G) λ (A)+1/ hece λ (A)=λ (G). Hece for N = G \ A, sice A is measurable, λ (N)=λ (G) λ (A)=0 ad A = G \ N. If λ (A) =, we set A m = A [m, m + 1) for m. By mootoicity, each λ (A m ) is fiite. For each m Iteger,, pick G (m) ope such that λ (G (m) ) λ (A)+1/2 +2 m +1. The, with G (m), N = G \ A is the desired set. m For the other directio, ote that the measurable sets form a σ-algebra which cotais both the Borel sets ad the ullsets. Hece ay set that is the differece of a Borel set ad a ullset is measurable, too. Oe ca also show that each Lebesgue measurable set ca be writte as a disjoit uio of a Σ 0 2 set ad a ullset. Hece if a set is measurable, it differs from a (rather simple) Borel set oly by a ullset. We also obtai the followig characterizatio of the σ-algebra of Lebesgue measurable sets. 7 5

6 Propositio 7.12: The σ-algebra of Lebesgue measurable sets i is the smallest σ-algebra cotaiig the ope sets ad the ullsets. As metioed before, there are Lebesgue measurable sets that are ot Borel sets. We will evetually ecouter such sets. The questio which sets exactly are Lebesgue measurable was oe of the major questios that drove the developmet of set theory, just like the questio which ucoutable sets have perfect subsets. Baire category The basic paradigm for smalless here is of topological ature. A set is small if it does ot look aythig like a ope set, ot eve uder closure. I the followig, let X be a Polish space. Defiitio 7.13: A set A X is owhere dese if its complemet cotais a ope, dese set. That meas for ay ope set U X we ca fid a subset V U such that V A. I other words, a owhere dese set is full of holes (Oxtoby). Examples of owhere dese sets are all fiite, or more geerally, all discrete subsets of a perfect Polish space, i.e. sets all whose poits are isolated. There are o-discrete owhere dese sets, such as {0} {1/: } i, eve ucoutable oes, such as the middle-third Cator set. The owhere dese sets form a ideal, but ot a σ-ideal: Every sigleto set is owhere dese, but there are coutable sets that are ot, such as the ratioals i. To obtai a σ-ideal, we close the owhere dese sets uder coutable uios. Defiitio 7.14: A set A X is meager or of first category if it is the coutable uio of owhere dese sets. No-meager sets are also called sets of secod category. Complemets of meager sets are called comeager or residual. The meager subsets of X form a σ-ideal. Examples of meager sets are all coutable sets, but there are ucoutable oes (Cator set). Baire category is ofte used i existece proofs: To show that a set with a certai property exists, oe shows that the set of poits ot havig the property. A famous example is Baach s proof of the existece of cotiuous, owhere 7 6

7 differetiable fuctios. For this to work, of course, we have to esure that the complemets of meagre sets are o-empty. Theorem 7.15 (Baire Category Theorem): For ay Polish space X, the followig statemets hold. (a) For every meager set M X, the complemet M is dese i X. (b) No ope set is meager. (c) If {D } is a coutable family of ope, dese sets, the D is dese. Proof. (a) Assume M = N, where each N is owhere dese. The M = D, where each D cotais a dese, ope set. Let U X be ope. We costruct a poit x U M by iductio. We ca fid a ope ball B 1 of radius < 1 such that B 1 U D 1, sice D 1 cotais a dese ope set. I the ext step, we use the same property of D 2 to fid a ope ball B 2 of radius < 1/2 whose closure is completely cotaied i B 1 D 2. Cotiuig iductively, we obtai a ested sequece of balls B of radius < 1/ such that B B 1 D. Let x be the ceter of B. The (x ) is a Cauchy sequece, so x = lim x exists i X. Sice for ay, all but fiitely may x i are i B, we have x B for all. Therefore, by costructio x B = B U D U. (b) follows immediately from (a), the proof of (c) is exactly the same as that for (a). I fact, the three statemets are equivalet. As a applicatio, we determie the exact locatio of i the Borel hierarchy of. Corollary 7.16: is ot a Π 0 2 set, hece a true Σ0 2 set. Proof. Note that caot be meager, by (b). Sice is meager, \ caot be meager either. If were a Π 0 2 set, it would be the itersectio of ope, dese sets ad hece its complemet \ would be meager. We have see that the measurable sets are precisely the oes that differ from a Π 0 2 set by a ullset. We ca itroduce a similar cocept for Baire category. 7 7

8 Defiitio 7.17: A set B X has the Baire property if there exists a ope set G ad a meager set M such that B G = M. The sets havig the Baire property form a σ-algebra ad hece iclude all Borel sets. Similar to measure, oe has Propositio 7.18: The σ-algebra of sets havig the Baire property is the smallest σ-algebra cotaiig all ope ad all meager sets. As i the case of measure, there exist o-borel sets with the Baire property, ad usig the Axiom of Choice oe ca show that there exists set that do ot have the Baire property. We coclude this lecture with a ote o the relatioship betwee measure ad category. From the results so far it seems that they behave quite similarly. This might lead to the cojecture that maybe they more or less coicide. This is ot so, i fact, they are quite orthogoal to each other, as the ext result shows. Propositio 7.19: The real umbers ca be partitioed ito two subsets, oe a Lebesgue ullset ad the other oe meager. Proof. Let (G ) be a sequece of ope sets witessig that is a ullset, i.e. each G is a uio of disjoit ope itervals that covers ad whose total legth does ot exceed 2. The G = G is a ullset, but at the same time it is a itersectio of ope dese sets, thus comeager, hece its complemet is meager. 7 8