Math212a1411 Lebesgue measure.

1 Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli lemmasMath212a1411 Lebesgue SternbergOctober 14, 2014 Shlomo SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli lemmasReminderNo class this ThursdayShlomo SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli lemmasToday s lecture will be devoted to Lebesgue measure, a creation ofHenri Lebesgue , in his thesis, one of the most famous theses in thehistory of SternbergMath212a1411 Lebesgue Lebesgue outer measure.

2 Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli lemmasHenri L on LebesgueBorn: 28 June 1875 in Beauvais, Oise, Picardie, FranceDied: 26 July 1941 in Paris, FranceShlomo SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli lemmasIn today s lecture we will discuss the concept ofmeasurabilityof asubset ofR. We will begin with Lebesgue s (1902) definition ofmeasurability, which is easy to understand and intuitive. We willthen give Caratheodory s (1914) definition of measurabiity which ishighly non-intuitive but has great technical advantage. For subsetsofRthese two definitions are equivalent (as we shall prove).

3 Butthe Caratheodory definition extends to many much more particular, the Caratheodory definition will prove useful for uslater, when we study Hausdorff SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli lemmasThe argument which will recur several times is: 112n= 1and so 1 2n= .We will call this the /2ntrick and not spell it out every time weuse SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli lemmas1 Lebesgue outer inner s definition of s definition of -fields, measures, and outer Borel-Cantelli lemmasShlomo SternbergMath212a1411 Lebesgue Lebesgue outer measure.

4 Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli lemmasThe definition of Lebesgue outer any subsetA Rwe define itsLebesgue outer measurebym (A) := inf `(In) :Inare intervals withA In.(1)Here the length`(I) of any intervalI= [a,b] isb awith thesame definition for half open intervals (a,b] or [a,b), or course ifa= andbis finite or + , or ifais finite andb= + , or ifa= andb= the length is SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli lemmasIt doesn t matter if the intervals are open, half open (A) := inf `(In) :Inare intervals withA In.

5 (1)The infimum in (1) is taken over all covers ofAby intervals. Bythe /2ntrick, by replacing eachIj= [aj,bj] by(aj /2j+1,bj+ /2j+1) we may assume that the infimum istaken over open intervals. (Equally well, we could use half openintervals of the form [a,b), for example.).Shlomo SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli lemmasMonotonicity, Bthenm (A) m (B) since any cover ofBby intervals isa cover any two setsAandBwe clearly havem (A B) m (A) +m (B).Shlomo SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures.]

6 The Borel-Cantelli lemmasSets of measure zero don t setZis said to be of ( Lebesgue ) measure zero it its Lebesgueouter measure is zero, if it can be covered by a countableunion of (open) intervals whose total length can be made as smallas we any set of measure zero, thenm (A Z) =m (A).Shlomo SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli lemmasThe outer measure of a finite interval is its [a,b] is an interval, then we can cover it by itself, som ([a,b]) b a,and hence the same is true for (a,b],[a,b), or (a,b). If the intervalis infinite, it clearly can not be covered by a set of intervals whosetotal length is finite, since if we lined them up with end pointstouching they could not cover an infinite interval.

7 We claim thatm (I) =`(I)(2)ifIis a finite SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli may assume thatI= [c,d] is a closed interval by what wehave already said, and that the minimization in (1) is with respectto a cover by open intervals. So what we must show is that if[c,d] i(ai,bi)thend c i(bi ai).We first apply Heine-Borel to replace the countable cover by afinite cover. (This only decreases the right hand side of precedinginequality.) So letnbe the number of elements in the SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity.

8 -fields, measures, and outer measures. The Borel-Cantelli lemmasWe need to prove that if[c,d] n i=1(ai,bi) thend c n i=1(bi ai).We do this this by induction onn. Ifn= 1 thena1<candb1>dso clearlyb1 a1>d thatn 2 and we know the result for all covers (of allintervals [c,d] ) with at mostn 1 intervals in the cover. If someinterval (ai,bi) is disjoint from [c,d] we may eliminate it from thecover, and then we are in the case ofn 1 intervals. So we mayassume that every (ai,bi) has non-empty intersection with [c,d].Shlomo SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli lemmasAmong the the intervals (ai,bi) there will be one for whichaitakeson the minimum possible value. By relabeling, we may assume thatthis is (a1,b1).

9 Sincecis covered, we must havea1<c. Ifb1>dthen (a1,b1) covers [c,d] and there is nothing further to do. Soassumeb1 d. We must haveb1>csince (a1,b1) [c,d]6= .Sinceb1 [c,d], at least one of the intervals (ai,bi),i>1contains the pointb1. By relabeling, we may assume that it is(a2,b2). But now we have a cover of [c,d] byn 1 intervals:[c,d] (a1,b2) n i=3(ai,bi).So by inductiond c (b2 a1) + ni=3(bi ai).Butb2 a1 (b2 a2) + (b1 a1) sincea2<b1. Shlomo SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli lemmasWe can use small repeat that the intervals used in (1) could be taken as open,closed or half open without changing the definition. If we takethem all to be half open, of the formIi= [ai,bi), we can writeeachIias a disjoint union of finite or countably many intervalseach of length<.]

10 So it makes no difference to the definition ifwe also require the`(Ii)< (3)in (1). We will see that when we pass to other types of measuresthis will make a SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity. -fields, measures, and outer measures. The Borel-Cantelli lemmasSummary of where we are so have verified, or can easily verify the following properties:1m ( ) = B m (A) m (B).3By the /2ntrick, for any finite or countable union we havem ( iAi) im (Ai).4If dist (A,B)>0 thenm (A B) =m (A) +m (B).5m (A) = inf{m (U) :U A,Uopen}.6 For an intervalm (I) =`(I).Shlomo SternbergMath212a1411 Lebesgue Lebesgue outer measure. Lebesgue inner measure. Lebesgue s definition of measurability. Caratheodory s definition of measurability. Countable additivity.