Measurable Functions and their Integrals
1Þ General measures: Section 10 in Billingsley
Recall: a probability measure T on a 5Y -field on a space H is a real- valued function on Y with the properties:
(a) TÐ9 Ñ œ ! (b) TÐH Ñ œ " _ _ (c) TÐ E333 Ñ œ TÐEÑ if E are disjoint. 3œ" 3œ"!
Definition 1: A measure T on Y satisfies the same conditions as a probability measure with the extension that condition (2) is eliminated, and sets may have infinite measure.
The measure .H is called 5-finite if is a countable union of sets of finite measure.
The combination ÐßHY Ñ is called a measurable spaceß and Ðß HY ßTÑ is called a measure space.
Example 1: Let Y‘! denote finite disjoint unions of finite intervals on , and 8 5YÐÑœœ! Y U = Borel sets on ‘ . As on Ò!ß"ÓEœN− , for 8! Y (with 3œ" N8 intervals), define 8 .ÐEÑ œ PÐN Ñß " 8 3œ" where P denotes length. Let .. also denote the unique extension of to YUœ [this is shown below].
[this works identically to how it's done on Ò!ß "Ó ]
Then .U on is called Lebesgue measure.
Note that .5 is -finite since ‘œ Ò3ß 3 "ÓÞ 3
71 8 Example 2: Let Y‘! be field of finite unions of rectangles in , i.e. sets 8 Eœ V8, where 3œ"
V8"833 œÖx œÐB ßáßB ÑÀB −N œinterval on ‘ ×Þ
8 Then Y5YUœÐ! Ñœ is called the Borel sets on ‘ Þ The measure on rectangles given by
.ÐV8"# Ñ œ PÐN Ñ † PÐN Ñ † á † PÐN 8 Ñ then extends to a measure on Y! by addition (i.e. measure of a disjoint union of rectangles is the sum of the measures of each rectangle), and then to a measure .U on by unique extension [see below].
Theorem 10.2 [Billingsley]: Let . be a measure on a measure space ÐßHY. ßÑÞThen (a) We have both continuity from above and below:if EÅE88 , then . ÐEÑ 8Ä_Ò.ÐEÑÞThe same holds if E8" Æ E , as long as . ÐE Ñ _Þ Ð,Ñ Subadditivity:
__ ..E Ÿ ÐE ÑÞ ."33 3œ" 3œ"
Theorem 10.3 [B]: If two 5..5Y -finite measures "#ßÐÑ on a field ! of sets in HY agree on !! , then they agree on 5YÐÑ . Moreover, this is also true if Y1! is only a -system and not a field.
[The proof of this follows directly from that for a probability measure space; see Billingsley]
2. Measurable functions: Billingsley, Section 13
Definition 2: Given two measurable spaces ÐßHY Ñ and Ð Hww ß Y Ñ , a map XÀHH Äww is called measurable if for every E− Y , we have XÐEÑ−Þ" Y
Example 2: Consider the function 0À‘‘ Ä given by Cœ0ÐBÑœBÞ# Then if EœMœ open interval œÐ+ß,Ñ , then if !+, ,
72 " ( 0 Ð+ß ,Ñ œÈÈ +ßÈÈ ,Ñ Ð +ß ,Ñ − U If +!,, then " 0Ð+ß,ÑœÐ,ß,Ñ−ÞÈÈ U Thus 0 " takes open intervals to sets in U
Now need theorem:
Theorem 13.1: If XÐEÑ" is measurable for all E−T (for some fixed collection T5TY ), and (Ñœw , then the map X above is measurable.
It is easy to show that if the set TU of open intervals generates , so above shows that 0ÐBÑ is measurable.
"Bif rational Example 3: Consider the function 0ÐBÑœ , on the real œ !Bif irrational Ú 9 if Ð+ß ,Ñ does not contain " or ! line. Then 0Ð+ß,Ñœ" if Ð+ß ,Ñ contains " and not 0 . Û ‘~ if Ð+ß ,Ñ contains ! and not " Ü But note that is measurable since it's a countable union of the rationals. Thus so is ‘µ0 , and is measurable.
$ ### Example 4: Consider 0 À‘‘ Ä given by 0ÐB"#$ ßB ßB ÑœB"#$ B B Þ " $ Then for example if +!, , 0 Ð+ß,Ñœ ball of radius È , in ‘ Þ Is this ball in U ? Clearly it's a countable union of small open cubes; thus it is in U and so by the above argument 0 is measurable.
Theorem 13.4: If 0À8 H‘ Ä are real-valued measurable functions, then: (a) sup0ß88 inf 0 , lim sup 0ß 8 lim inf 0ß 8 are measureable. 8888 (b) If lim 08 exists everywhere then it is measurable. 8Ä_
3. Integral of a function
Given a function 0ÐBÑ on Ò+ß,Ó , we define its integral 0ÐBÑ... (with say ' Lebesgue measure) as the area underneath it:
73 We can view this as a sum of small rectangles in approximation:
To find the area we add the areas of rectangles under the curve by computing the limit 8 lim 0ÐB33 Ñ. ÐE Ñß 8Ä_" 3œ" where EœÐBßBÓÞ33"3
Basically, we are decomposing the domain Ò+ß ,Ó into measurable sets E3 and finding the size of the function on each E3 to get the area of the rectangle above E3 which is still below the function.
74 For a characteristic function "B−Eif 0ÐBÑœ ME ÐBÑœ œ ! otherwise this is easy: we have
MÐBÑ.BœÐEÑÞE . ( By extension, if we have a sum of characteristic functions with disjoint supports such as 8 0ÐBÑ œ - M ÐBÑ, " 3 E3 3œ" [we call these simple functions ] we similarly define 8 0ÐBÑ.B œ -. ÐE ÑÞ " 33 ( 3œ" We can then integrate any positive function which is measurable on ‘ via the following theorem. Note it also works exactly the same way if we replace ‘H by any measure space :
Proposition 4: Given any non-negative measurable function 0 on H , there is a sequence 00Å088 of simple functions such that . 8Ä_
Proof: Let
777" ##888if 0ÐBÑ− Ò2 ß Ñ and 0ÐBÑ 8 0ÐBÑœ8 . œ 80ÐBÑ 8if Then clearly at any B80ÐBÑ for sufficiently large , if is finite we have " ! Ÿ 0ÐBÑ088 ÐBÑŸ#8 if 0ÐBÑ is finite, so that 0 ÐBÑ8Ä_Ò 0ÐBÑÞ If 0ÐBÑœ_, then also 08 ÐBÑœ88Ä_Ò 0ÐBÑœ_Þ It is also easy to show that 0ÐBÑ 0ÐBÑ8" 8 .
7 Then letting E87 œÖBÀ0 8 ÐBÑœ28 × , and E 8! œÖBÀ0ÐBÑ 8× 8 , we have
75 8 7 0ÐBÑœ M 8M , 8 " 8 EE87 8! 7œ" # proving the proposition.
5. Properties of the integral of a non-negative simple function
Definition 3: A statement about a measure space H is true almost everywhere ( a.e. ) if it is true everywhere in H except on a set of measure ! .
Theorem 5: If 01 and are non-negative simple functions, then (a) If 0Ÿ1 a.e., then ... 0Ÿ . 1 '' (b) .01œ.0.1.! " ! . " . '''ab (c) If 0œ! a.e., then .. 0œ! '
Proof: If 0 œ 0ÐE33F ÑME3 and 1 œ 1ÐF ÑM3 , we note that both 0 and 1 !!33 are constant on the finite collection of sets ÖE343ß4 F × . Thus without loss henceforth we will assume that 01 and are constant on the same collection of sets, without loss called ÖE3 × .
To prove (a), note clearly that if
0 œ 0ÐE ÑM and 1 œ 1ÐE ÑM , Ð"Ñ ""33EE33 33 then clearly .0œ0ÐEÑÐEÑŸ1ÐEÑÐEÑœ.1ß..33 33 .. since ''!!33
0ÐEÑ33 Ÿ 1ÐEÑ for all EE33 with positive measure, and since of measure 0 do not contribute to Ð"ÑÞ
Proving (b) is also simple using the representation (1) above.
4. Definition of the integral of a positive function
76 Definition 4. Given a positive function 0 on H , we define its integral to be
0ÐBÑ... œlim 08 ÐBÑ. ,(1) ((HH8Ä_ where 0ÐBÑÅ8 0ÐBÑ is a sequence of simple functions (note Å always 8Ä_ means the sequence is nondecreasing or monotonic , so we are allowing two successive terms to be equal.Ñ
Note this limit exists since it is easy to show that the sequence .0ÐBÑ. is ' 8 increasingß0 since 8 are.
Proposition 6: The integral defined in (1) is well-defined. That is, if 08 and ˜ 008 are simple function sequences converging to monotonically, then the integrals defined by these sequences are identical to each otherÞ
We may assume that 0Á!8 on a set of finite measure (and similarly for 0ј ; otherwise it is easy to show ... 0 œ . 0œ_ , and the proof is 8 ''8 simple.
˜ Let 008 and 8 be two monotonic simple sequences converging to 0 . It suffices to show that for any 8!" , there is an 8! such that (no matter how small $ is)
˜ .0 Ð"Ñ.$.$8 .08" ß (1a) ((HH ˜ along with the same statement with 008 and 8 reversed. This would show (if $.. is small) that .0˜ and .0 come arbitrarily close to each other for ''8 8" arbitrarily large 88" and , meaning that their limits must be the same.
To prove (1a), note that for fixed 8" and fixed $ , ˜ EœÖBÀ0Ð"Ñ0×Æ888 $9" (empty set), 8Ä_ ˜ since 08 Å 0 , and 0 08""" Þ (Note . ÐEÑ_ , since 0 8 ÐBÑÁ! on a set of 8Ä_ finite measure). Thus for 8 sufficiently large, we have
77 ˜˜ .0..88 . 0 Ð"Ñ $.$.. .0888""" œÐ"Ñ . 0 .0 -- ((EE88 (Œ ( ( E8
Ð"$.. Ñ . 0888 ÐEÑmax 0 Ð" $.$ Ñ . 0 8 Œ(("" " with the last inequality following since .ÐE8 Ñ gets arbitrarily small as
8Ä_, while max 08" and $ are fixed and finite. Statement (1) is thus proved, as desired.
5. Alternative definition of an integral
Definition 5 (Billingsley): We can also define the integral of a non- negative function 0 by the right side of equation (1) below, which is Billingsley's definition.
The following theorem shows that our definition is equivalent to Billingsley's:
Proposition 7: Given a measure space ÐßHY. ßÑ and a non-negative measurable function 0 on H ,
.0œW´.=.sup inf 0ÐÑ ÐEÑ3 (1) " =−E (H 3 Œ3 where the sup is over all possible partitions ÖE3 × of H into a finite number of measurable sets.
Note that W above is defined to be the right side of (1).
Proof: Let 08 be an increasing (i.e., non-decreasing) sequence of simple functions converging to 0Þ Clearly we have Ð since 0883 Ÿ 0 on E Ñ
... 0 œlim 088383 ÐE Ñ ÐE Ñ Ÿ lim sup inf 0Ð =. Ñ ÐE 83 Ñ Ÿ Wß 8Ä_""=−E ( 338Ä_ Œ83 where 0ÐEÑ883 is the (constant) value of 0 8 on the set E 83 Þ
78 On the other hand, by the definition of W , there must be a sequence of partitions ÖE83 × 3 for each 8 such that
Wœlim inf 0ÐÑ=. ÐEÑ83 8Ä_ " =−E 3 Œ83
Now define the sequence 08 of simple functions such that 0ÐE83 Ñ œinf 0Ð= Ñ, so that 0 8 Ÿ 0Þ Then =−E83
Wœlim 0ÐEÑ88383.. ÐEÑŸ . 0ß 8Ä_" 3 ( which proves (1).
6. The integral of a general function
Now for any function 0ÐBÑ , define 0ÐBÑif 0ÐBÑ ! 0ÐBÑœ max Ð0ÐBÑß!Ñœ . œ ! otherwise
0ÐBÑif 0ÐBÑ ! 0 ÐBÑ œmax Ð0ÐBÑß !Ñ œ . œ ! otherwise
Then any function 0ÐBÑ can be written as 0ÐBÑœ 0 ÐBÑ0 ÐBÑ . We then define
.0œ.. .0 .0Þ . (1c) (( ( Thus we have defined an integral for all measurable functions 0ÐBÑ .
Definition 6: The function 0 is integrable if both integrals on the right of (1c) are finite.
6. Infinite values
Definition 7: We add the numbers __ and to ‘ as the extended real number system, with the algebra rules
79 +_œ_ if +Á_
+_œ_ if +Á_
__ is undefined Henceforth we will assume our functions are extended real valued. The above discussion and theorems are unchanged with the addition of infinite values for functions.
For Proposition 4 above, if 0ÐBÑœ _ for B − E œ set of positive measure, then it's easy to show that .0..Ò and .0˜ _ , so that it still holds ''8 8 8Ä_ true. Similarly for Proposition 5, in that case it's easy to show directly that both sides are _ , so that the equality still holds.
Note however that values of simple functions will always be assumed finite by definition.
Proposition: If .0_. then 0_ a.e. 'H Proof: Otherwise there would be a set EÐEÑ!0œ_ with . on which . Then there is a sequence of simple functions 0Å08 which are constant and 8Ä_ equal to _ on E , so that their integrals ... 0 œ_ , and so . 0œ_ . ''8 7. Properties of the general integral
Theorem 8. The standard properties of the integral of simple functions (Theorem 3) follow also for general functions 0 , i.e., (a) If 0Ÿ1 everywhere, ... 0Ÿ . 1 '' (b) .01œ.0.1.! " ! . " . '''ab (c) If 0œ! almost everywhere, then .. 0œ! ' This theorem follows directly from Theorem 3 by approximating 01 and by simple sequences 0Å088 and 1Å1 . 8Ä_ 8Ä_
8. Integration to the limit
80 Theorem 16.2 (Monotone convergence theorem): If !Ÿ08 Å0 almost everywhere, then
.0ÐBÑÅ..8 .0ÐBÑ ((HH Proof: Without loss of generality we may restrict H to those points where 0Å08 , and so assume that convergence occurs everywhere (since integrals over sets of measure 0 are 0 anyway). Clearly by Theorem 5, .0. are ' 8 increasing. Let 28 be simple functions defined by 777" ###888if 0ÐBÑ−Ò88 ß Ñ and 0ÐBÑ8 2ÐBÑœ8 œ ! otherwise
Then if B is such that 08 ÐBÑ8Ä_Ò 0ÐBÑ _ , then for sufficiently large 8 , " we have 088 ÐBÑ 2 ÐBÑ Ÿ#8 , and so 2 8 ÐBÑ8Ä_Ò 0ÐBÑÞ Similarly it is easy to show that if 0ÐBÑœ _ , then 28 ÐBÑ8Ä_Ò _Þ Thus
lim ... 28 ÐBÑ œ . 0ÐBÑß (2a) 8Ä_(( while
.2Ÿ...88 .0Ÿ .0ß (2b) ((( which implies by (2a,b) that
lim .0œ..8 .0Þ 8Ä_((
Theorem 16.3 (Fatou's Lemma): For non-negative 08 ,
...lim inf 088 ÐBÑ Ÿ lim inf . 0 ÐBÑÞ ((HH88 [simple proof in Billingsley]
Theorem 16. 3 (Lebesgue Dominated Convergence Theorem): If l088 l Ÿ 1 a.e. and 1 is integrable, then if 0 converge a.e.,
...lim 088 ÐBÑ œ lim . 0 ÐBÑÞ ((HH8Ä_ 8Ä_
81 [proof in Billingsley].
9. Integration over sets
Recall we can write the characteristic function "B−Eif MEE ÐBÑ œ; ÐBÑ œ . œ ! otherwise
Note we can write
"_ B## B /.BœMÒ"ß"Ó /.B; ((" _ more generally we define for measurable E§H
0ÐBÑ... œ ME ÐBÑ0ÐBÑ. . ((E H This defines integral over a set E ; easy to show that all properties of integrals carry over when the space H is replaced by the subset E .
82