125

APPENDIX

THE LIFTING THEOREM

In this appendix we give a simple direct proof of the lifting theorem,

in the case of a finite measure, which clearly brings out its connection

with differentiation. The methods we follow are close in spirit to

those originally used by yon Neumann working with Lebesgue measure on

the real line. For a good guide to the literature on this subject, we

refer the reader to the book by A. and C. lonescu Tulcea, 'Topics in the

Theory of Lifting', Springer-Verlag, 1969.

We assume only acquaintance with the basic notions of Caratheodory outer

measures on an abstract space, thereby making this appendix essentially

self contained.

O.

126

Basic Notation

S

~+=

60 =

Throughout this appendix,

is an abstract space;

is a non-negative outer measure on S with %(S) < ~;

is the family of %-measurable sets;

{A ¢~: 0 < %(A) < %(S)} o {~, S};

{A: %(A) = 0};

{0, i, 2, }.

For any subsets A, B of S,

A \ B = {x e A: x / B};

~.B= S\B;

A ~ B iff %((A\ B) O (B~ A)) = 0.

For any family F of subsets of S,

OF= U ~ , cc~F

B o r e l f i e l d F = s m a l l e s t o - f i e t d c o n t a i n i n g F;

B F = B o r e l f i e l d ( F u ~ )

( = c o m p l e t i o n o f B o r e l f i e l d F w i t h r e s p e c t t o ) t ) ;

)~F = The o u t e r m e a s u r e g e n e r a t e d by X/ (F u ~ ) , i . e . f o r any A c S,

% F(A) = inf { E X(~); H is countable, H~F , A~UH}; ash

F is complete iff, for every A~ S and B ~ F, if A ~ B then A ~ F.

127

Liftings

Definitions

For any field BC~,

(I) p is a density function on 8 iff p : B ÷~+

and, for every A,B ~ B,

(i) A e p(A)

(ii) A z B ~--~p(A) = p(B)

(iii) p(Ao B) = o(A) ~ o(B)

Note that (i) and the definition of~ + imply

p(~) = ~ and p(S) = S.

(2) p is a lifting on ~ iff p is a density function on

which satisfies the added condition

(iv) p(~A) = up(A) for A ~ 9.

(3) F lifts B iff FcA +, F is a field and, for every

A e B, there is a (necessarily unique) ~ s F with A

(4) F is a partial liftin~ iff F lifts B F.

Note: If p is a lifting on ~ then range p lifts 9.

Conversely, if F lifts B and, for every A s B, p(A)

is the ~ e F with A ~ ~ then p is a lifting on B.

128

2. Differentiation Systems

Definitions

(i) F is a differentiation system iff F=~+ and, for

every ~,B ~ F, we have ~ ~ B ~ F.

(2) For any differentiation system F and s e S,

~(s) = {~ ~ F: s E ~}

and F(s) is directed by inclusion downward.

(3) For any differentiation system F, H~F and A~S,

H is a Vitali cover for A iff, for every s ~ A and

B e F(s), there exists ~ ~ H with s e ~ ~-B.

(4) F is a Vitali system iff F is a differentiation system

such that

(i) for every A~S and Vitali cover H for A, there

exists a countable, disjoint H' ~ H with

~(A\ OH' ) = O;

( i i ) ~F / BF = ~ / BF"

129

3. Main Steps to the Lifting Theorem

Here we only list the main steps leading to the lifting theorem. Their

proofs are given in the next section.

Theorem i If F is a partial lifting then F is a Vitali system.

Theorem 2 If F is a Vitali system, T~- S, r ~ 0,

lim l(= ~ T) ! D, = {s s S : ~sF-~s) ~ (=) r },

lim %(= ~ T) D* = {s e S : =sF(s) ---% (=) r },

then D, E BF, D* s B F and, for any B e BF,

I(B ~ D, tl T) r • %(B ~ D*).

Theorem 3 If, for each n e ~, F n is a Vitali system, Fn~ Fn+l,

A e Borelfield (nEll B F ) , and n

B = {s g S : lim lim %(= ~ A) = i} -^ n =eFn(S) X (=)

then A ~ B.

Theorem 4

Theorem 5

If, for each n c ~, F n is a partial lifting and F n Fn+l

then there exists a density function p on Borelfield (UBF) ne~ n

with p(A) = A for A E U F . new n

If p is a density function on a complete field B =~ then

there exists an F which lifts B and such that

(A) ~A~up(~A) for A e F.

Corollary

Lemma

If, for each n c ~, F is a partial lifting and FnCFn+ I n

then there exists a partial lifting H with U F ~H. ns~ n

If F is a partial lifting and A e ~%B F then there exists

130

Liftin$ Theorem

a partial lifting F' such that F c-F' and A ~ BF,.

There exists a maximal partial lifting F with B F =~.

Thus, there exists a lifting on ~.

131

4. Proofs

Here we give the proofs of the theorems listed in the previous section.

In fact, we prove somewhat stronger results, for the sake of perspective.

Theorem 1 If F is a partial lifting then F is a Vitali system.

Proof: (i) Given A ~S and a Vitali cover H for A, let H' be a maximal,

disjoint subfamily of H. Then H' is countable and there

exists $ ~ F with (S ~ OH') - ~. For every = s H', we

have ~ ~ ~ ~i ~ and ~ A ~ ~ Fc ~+ hence ~ ~ ~ = ~. There-

fore, we cannot have any s c A f~ B for, otherwise, there

would be an=' ~ H with s ~ =' C B contradicting the maxi-

mality of H'. Thus, A ~ B = ~ and so %(A ~- UH') = 0.

(ii) Since F lifts B F clearly %F / BF = % / BF"

Theorem 2' Let F be a Vitali system, r _> 0, v be a finite measure on

B F which is absolutely continuous with respect to %/BF, lim v(~) < r },

D. = {s c S : ~--Y(~(s) ~(~) -

lim v(=) > r }. D* = {s ~ S : ~E~(s) ~(~) --

If v is the outer measure generated by v then

(I) A = D, => ~(A) ](A) > r.% F (A)

(3) D, g B F and D* E B F-

Proof: For any r' > r and B e F, let o

H = {= s F : = ~ B and ~(=)< r'} ~(~)

Then H is a Vitali cover for B ~ D,, hence there exists a

countable, disjoint H' C H with %(B ~D,~ UH') - 0°

So v(B ~ D, \ OH') = 0 and v(B ~ D, ~ OH') = 0 and

132

(1)

(2)

7(B ~ D,) j Z v(=) ~ r' Z %(=) ! r' . ~(B). =EH v ~EH v

given any A~S and e>O, from the definitions of %F and

and absolute continuity of ~, there exists B e F such that o

X(A \B) = 0 so ~(A N B) = 0 and ~(A N B) = O, and

%(B) i %F (A) + ~"

If A C D, then

](A) i ](B m D,) i r' • %(B) ! r' • %F(A) + r' • ~.

Letting r' ÷ r and ~ ÷ O, we get

A CD, =>](A) ! r • %F(A)

Similarly, we get

A C D* => ~(A) ~ r . %F(A).

To check (3), let B be a common %F' ~ - outer hull of D,,

i.e., let B e BF, D, CB, %F(D,) = %(B) and ~(D,) = v(B).

Then, for every = ~ B F with = ~B, we have

%(~) = XF(= ~ D,) and ~(~) = ~(~ ~ D,).

To see that %(B ~ D,) = 0. let

A = {s ~ B~D, : lim ~(=) 1 n =~F(s) ~(=) > r +--n }

and choose =n E B F so that An~ =n~ B,

%(=n ) = kF(An) and V(=n) = ~(An). Then, by (2),

](A n) > (r + i) . %F(An)

On the other hand, with the help of (i), we have

](An) = V(=n ) = ](=n ~ D,) ! r • %F(=n ~ D,) = r • %(~n )

= r • IF(A). Therefore . XF(An) = 0 and, s i nc e B \ D . ~ n ~ An, we conclude

133

Theorem 2

Theorem 3

Proof:

X(B ~ D,) = 0 so D, s B F. Similarly, D s B F.

is then an immediate corollary of theorem 2' obtained by

letting ~(A) = I(A ~ T) for A e B F.

If, for each nsm, F is a Vitali system, F ~ F n n n+l'

B = B o r e l f i e t d (n~g] B F ) , A s B, and n

B = {s e S : lim lim I(= ~ A) = i} n ~eFn(S) l(~)

then A ~ B.

For any r < i, let

D,(n) = {s e S : lim X(= ~ A) < r},

n

E, = E,(r) = /~ O NEe n>N D,(n).

Then (A\ B)~ U~ ~ E,(r)). r N,

n-i D ! = ° n D,(n) ~ U D,(j)

j=N

Then $ /% D'n E B F n

and so by theorem 2,

X(fl ~ D' /% A) < r • X(fl ~ D' ). n -- n

Summing over n h N, we get

k(B/% n>~N D,(n) ~ A) ~N D,(n)),

hence

%(B /% E, /~ A) < r • %(~ /% E,).

By considering monotone sequences of such B's, we conclude

134

Theorem 4

Proof:

that the above inequality holds for any $ ~ ~. In parti-

cular then, letting B = A, we get I(A ~ E,) ! r • I(A ~ E,)

and, since r < i, we must have I(A ~ E,) = O. Letting r + 1

through an increasing sequence, we conclude X(A \ B) = 0.

Similarly, for any r > O, if

D*(n) = {s s S : lim ~(= ~ A) > r},

~n(S) x(~)

E~ = f~ n>UN D~(n) N~W

and ~ a 8 then

~(~ m E*m A) > r • ~(S m E*).

Letting B = ~A, we get I(E*~ A) = 0 for every r > 0, so

for ~ - a.a. s a (S \ A),

lim lim %(= ~ A) = O. n ~E~ (s) ~(~)

n

Thus, %(B \ A) = 0 and A = B.

If, for each new, Fn is a partial lifting, Fn = Fn+l and

~ = B o r e l f i e l d (nYw BF ) t h e n t h e r e e x i s t s a d e n s i t y f u n c t i o n n

p on B with p(A) = A for A ~ (.7 F . new n

For every A ~ B, let

p(A) = {s ~ S : lim lim ~(~ ~ A) = i}. n ~EF (s) %(e)

n

By theorem 3, A E o(A) and, since each F is a field,

A c U F =>p(A) = A. new n

For any A, B E B, we trivially have

A ~ B =>p(A) = p(B).

Finally, to see that p(A ~ B) = p(A) ~ p(B),

135

(i)

(ii)

(iii)

let

f A(s) = lim %(~ ~ A) n esF (s) %(~)

n

A(s) = lim %(a ~ A) n eeF (s) %(~)

n

gA(s) = lim f A(s) n

n

gA(s) = lim f A(s) n

n

and note:

s s 0(A) gA(s) = i.

A~B AnB gA g (s) = 1 => i = g (s) g(A U B)(s) = I

=> 0 gB(s) - g(B \ A)(s) =

=> g(A ~ B)(s ) = i

so p(A) f% p(B) C P(A /%B).

Theorem 5

Proof:

(i)

(ii)

(i)

(2)

(3)

(4)

(5)

136

If p is a density function on a complete field B c ~ then

there exists an F which lifts ~ and such that

p(A) CA ~up(~A) for A s F.

Let p*(A) = up(~A). Then, for any A,B e B,

A --- p*(A)

p*(A v B) = p*(A) v ~*(B)

since ~A - o(~A) and p(~A f% ~B) = o(~A) ~ p(~B).

Let

A = {F : F is a field, F ~B and, for every = s F,

p(=) ~ = .- p*(~)}

and F be a maximal element in A. To check that F lifts B,

given A g 8, let

= ~ [p(~ u A)\ ~]. A'

We shall show that A -= A' and A' e F. First note:

=, fl E F => [p(= U A) X ~] ~ [P(B V ~A) X B] = $

for, P(~ V A) ~ p(B V ~A) ~ ~ ~ ~B ~ O(= u B) f% ~(= V B) = $.

o(A) CA' ~p*(A) so by (i) A = A'

for, @(A)~A' from definition of A' since $ ¢ F~

A' C o*(A) from (2) with B = ¢.

8 ¢ F => B~ A' Cp*(B /%A')

for, from (3) and (2), p(~B ~A') = p(~B U~A) C~B o ~A'.

y s F => y ~ A' ~p*(y ~ A')

for, p(~ O A') = p(~y u A) ~ ~y U A'.

Thus, if F' is the field generated by F ~ {A'} so that

F' = {~ U (S ~A') U (y ,,A'); ~, ~, y c F}

137

Lemma

Proof:

Liftin$ Theorem

then, by (ii), (4) and (5), for any C c F', we have

C ~p*(C) and ~C~o*(~C) hence p(C) CC ~p*(C).

Thus, F C F' c A and therefore F = F' and A' s F.

If F is a partial lifting and A c (~\BF) , then there

exists a partial lifting F' with F CF' and A ~ BF,.

Let

F 1 = {= c F : (= ~ A) - ¢}

F 2 = {= e F : (~ ~ A) ~ ~}

then choose B I c (FI) ~ and B 2 s (F2) ~ with

X(BI) = sup {%(=) ; = c F I}

%(B2) = sup {%(=) ; = s F 2}

and

~i, B2 ~ F with B 1 z B1 and B 2 - B2-

We must have B1 c FI, 82 c F2,

UFI = ~I since = c F I => %(= ~ ~I) = 0 => (= ~ BI) =

~F2 = B2 since = c F 2 => X(= "~ B2) = 0 => (= "~ B2) = ¢

B I f~ B 2 = ~ since (~i f~ B2) ~ (BI ~ A) U (B2 ~ A) - ¢

Let A' = (A \ BI) U B2 so A -= A' and

F' = {= u (B ~ A') u (y ~A'); =, B, Y e F}.

Then F' is a field, F~F' and F'C~ since, for B, y e F:

(8 ~ A') ~ ~ => B c F I => ~ C 81 => ~ /I A' =

(y ~ A') ~ ~ => y e F 2 => y = $2 => Y ~ A' = ~.

Since BF, = {~ U (B O A) ~ (y ~ A); =, B, Y ~ ~F

we see that F' lifts 8F' and A c ~F'"

There exists a lifting on

138

Proof:

(1)

Let~fbe a maximal nest of partial liftings and L = U~.

Then L is a partial lifting, for

If ~has no cofinal sequence then

(2)

B L = F~BF

so L lifts B L-

If~has a cofinal sequence then, by the corollary to

theorems 4 and 5, there is a partial lifting F' with L CF'

and hence L = F'.

Since L is a maximal partial lifting, by the lemma we must

have ~= B L. Thus, L lifts ~. If, for A g~, p(A) is the

s L with A £ = then p is a lifting on ~.

Remarks

(i) Theorem i above is a special case of theorem 3.3 in Chapter III.

The historical remarks at the end of Chapter III (p.123) apply

here too.

(2) Theorem 2 is a well known classical result whereas theorem 3

is a simple version of the Martingale theorem.

(3) The material in this appendix is taken from my Lecture Notes

"A Proof of the Lifting Theorem", University of British Columbia,

March 1970. I am indebted to J. Kupka for pointing out minor

errors in these notes and suggesting a way to avoid the use of

universal subnets in the proof of theorem 4. The proof given

here is partially based on his suggestions.

139

5. Lifting and Differentiation

The essence of the relation between the notions of lifting and differ-

entiation is contained in theorem i. The nature of the relation becomes

clearer if one realizes that theorem 2' is a key step in the Lebesgue

approach to differentiation, for one readily concludes from it that, if

f(s) = lim ~(=) ~ ( s ) x(~) '

then f(s) exists for X-almost all s ~ S, f is a BF-measurable function

and, for every A e BF,

v(A) = f f d%. A

With the ~elp of this, theorem 1 can be sharpened to yield:

Theorem I'

(i)

(ii)

Proof:

(i)

(ii)

If p is a density function on a complete Tfield B

and F = range p, then F is a Vitali system with B F = B.

If F is a Vitali system then there exists a density

function p on B F with A~p(A) for every A g F.

Same as theorem i, since the fact that A g F => ~A ~ F was

not used there.

Let

p(A) = {s E S : lim X(= m A) = i}

Theorem i"

(i)

(ii)

Theorem

140

If F lifts a complete field B then F is a Vitali system

with B F = B.

If F is a Vitali system then there exists an F' which

lifts B F .

The lifting theorem then can be restated in the form

There exists a Vitali system F with ~= B F.

This points out the significance of the lifting theorem for differenti-

ation: it provides us with a Vitali system in a general measure space,

to play the role of the family of intervals on the line, and thereby

enables us to follow the classical Lebesgue approach in differentiating

a measure ~ with respect to % to obtain an integral representation for

~. What makes this approach very useful is that, unlike the situation

with the Radon-Nikodym theorem, the Vitali system produced here depends

only on the base measure % and not on the measure ~ being differentiated.

As a result, by this approach one can obtain integral representations

for a vector-valued measure v even when v has, in an essential way, an

uncountable number of coordinates.