...
.....
..
- ,!!!! 1
-111 - M- ! ,
-
, -
-- -----
-6
- -
I-
- I
~~~~1
THE LEBESGIJE DENSITY THEOREM
IN ABSTRACT MEASURE SPACES
by
MIRIAM AMALIE LIPSCHUTZ YEVICK
B.A., New York University,
M.S.,
1943
Massachusetts Institute of Technology, 1945
Submitted in Partial Fulfillment of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY
from the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
1947
Signature of Author..,
Department of Mathematics,
September 5, 1947
Signature of Professor
in Charge of Research....
Signature of Chairman of Department'
Committee on Graduate Studies....
USQ
U
TABLE OF CONTENTS
Page
Acknowledgements
I
Introduction and Summary of Results
Chapter I:
II
The Density Theorem in Euclidean Space
Chapter II:
Chapter III:
Chapter IV:
Appendix: I:
The Density Theorem in Abstract Measure
Spaces
The Density Theorem in Finite Product
Measure Spaces
The Density Theorem in Infinite
Product Measure Spaces
Proof that
m(S(Xn))
O
J-
II: Procedure used in the construction of
the example of Chapter IV
Biographical Note
289911
1
6
14
18
28
30
_
ACKNOWLEDGEILETS
The author wishes to express her gratitude to Professor
Witold Hurewics for suggesting the problem of this thesis
and for his kind help and encouragement during the time that
this work was in progress.
INTRODUCTION AND SUMMARY OF RESULTS
Let us consider a measurable set
define:
a) The average density of
C-4 1);
b) The density of
(24
lim
-;
-4
O
A
A
A
on the line.
in an interval
at a point
P
We
I as
as
where the sequence of intervals
In
C.,,]
'i
In = d(In) -,O,
converges to
P.
for all
The Lebesgue density theorem asserts that the
n.)
(i.e. diameter
A = 1
density of any measurable- set
and equals nero for almost all
In>P
for almost all
PF CA.
P e A
This theorem can
be interpreted as stating that generally a set is not uniformly distributed over the line byt rather that its points
are very concentrated in certain regions and very rare in
others;
the exact measure of the concentration is specified
by the inequality:
The set of intervals 6c (A) in which the average den.sity of A
0)
exceeds
a., 0
a
1, has measure
.-2 ~ )
)) 6z CO
/1
m
If we are given in an abstract measure space
CA)
E, a family
of measurable sets (which we shall call elementary sets)
and a notion of convergence, and
point
p
of
E
S.
E
a concept of density with respect
We now assume in
inequality similar to (1).
existence of a function qf6
L
is such that to each
converges at least one sequence of sets of
S, we can define in
to the family
S
E
for this family an
Specifically we assume the
=
l.u.b.
(/2(p),such that the
S
liii
union6,: fA) of elementqry sets in which
density of
A
exceeds
satisfies:
a
A
9!
C4)
(2)
the average
(
4
Busemann and Feller (B3) hage shown that in Euclidean spaces
the validity of (2) for a family of sets
S
implies that
S.
the density theorem holds with respect to the family
Theorem I of Chapter II states that this sufficiency condition can be extended to abstract spaces provided we
impose upon the sets of
S
the restriction that given an
arbitrary system of elementary sets
S' C S
any measurable
set can be approximated in measure to withing any E> 0,
a sequence of sets
Bn, union of sets of
S'.
by
From the
validity of the density theorem with respect to a family
of sets follows, that the Lebesgue integral of any bounded
function can be differentiated with respect to this family
and that the derivative of the integral equals the integrand.
Theorem I of Chapter III states that if (2) holds in a
finite number of abstract spaces
functions
-
with
96(d} then for any measurable set
the direct product space
when'W'r/
El, E2,.---. *n
A
in
E =7T Ei
is the union of all elementary sets in
El
(i.e. all sets which are the product of elementary sets in
the fadtor spaces) in which the density of
A
It follows in particular that since (2) holds in
the density theorem holds in
a.
exceeds
R1,
Rn for the family of all inter-
vals and that differentiation with respect to this family is
allowed.
Proceeding to the product of an infinite number of
abstract spaces, Theorem I of Chapter IV asserts that condition (2) breaks down; i.e. for a bounded set
6 , (A)
A
the set
may be arbitrarily large.
Moreover an example is constructed in Chapter IV
leading to Theorem II which states that:
The density
theorem is not valid with respect to the set of all intervals in the space
interval.
E
=7T
Ei
whose factors are the unit
The same can be expected to be true for the
direct product of an infinite number of arbitrary finite
abstract measure spaces unless the elementary sets and
the functionsc4(V)are greatly restricted.*
From this follows theorem III of Chapter IV stating
that in differentiating an integral in an infinite number
of dimensions as defined for instance by Jessen, we are
not entitled to performing the limiting process involved
consecutively with respect to each variable separately.
Thus the main results of this thesis are contained in
1, i.e. the only
*For instance if 4t() is always equal to
elementary sets converging to points of
itself.
A
are the set
A
Theorems I of Chapters II and IIt, and Theorems I, II, and
III of Chapter IV.
Finally we note that Theorem I, Chapter II, can also be
considered as a theorem on conditional probabilities.
if the spaces
m(A)
Ei
For
consist of a finite number of points and
is ~defined as the ratio of the number of points in
to the total number of points in
conditional probability of
A
E, then '4
with respect to
4T) is
A
the
I. We
believe that in this form Theorem I of Chapters III and IV
should possibly allow of some interesting applications in
probability theory and statistics.
I
I.
THE DENSITY THEOREM IN EUCLIDEAN SPACE
In the extension of the Lebesgue density theorem to
higher dimensions a difference arises between differentiation with respect to the system of all cubes or any system
regular with respect to the set of cubes and differentiation with respect to systems that do not satisfy the
regularity requirement.
An absolutely continuous set function, i.e. an
F (e)
indefinite Lebesgue integral
41P can be
differentiated with respect to any system regular with
respect to the set of all cubes and the derivative of
F(p) = f(p).
A family of sets
with respect to the family
P
R
T
is said to be regular
of cubes, if
of the space converges a sequence (
is contained in a cube
each
__P)
o
;__
of
to each point
T
R
such that
with for all
k
-where o(P) is independent of
the particular sequence chosen.
Letting
E
f(p)
be the characteristic function of a set
it follows that the density theorem holds with respect
to any such system.
A simple geometrical proof of this
fact in which the &are
by Sierpinski (S2).
n
dimensional spheres was given
The same result was obtained by
de la Vallee Poussin (V2) using the notion of differentiation on a net.*
*This notion becomes very useful when considering infinite
Euclidean product Spaces.
SIM
These proofs and the proof usually given (see
Caratheodory (Cl) ) are esaentially based on the following
lemma.
Lemma I.
A
Let
A
be a set such that in each point
for any sequence 4
. (~P
/~
>
converging to
A
O(
p
of
p,
?~and the same with the
inequalities reversed.
This lemma can then be combined with the following
(see R. de Possel (P2) ).
Lemma II.
If
g(p) are two functions defined
and
f(p)
in the points of a space
R, such that whenever
in all the points of a set A, with
a point
all
p
p
of
of
A
with
f(p)-7 a
g(p) > a
m(A)2> 0, there exists
then
f(p)-P. g(p)
for
R( and identically with the inequality signs
reversed)
and therefore
RiP) >o<
,
3
JF
1(y)
-,)(P)hs
>
and
')
reversing the inequalities
C.e.
d
()>o(implies
We note that sinceJ2
=
The proof of Lemma I is based directly on Vitali's covering
theorem.
Namely, we cover the set
number of disjoint sets
A
7
9
i
with an enumerable
in each of which
From the complete additivity and absolute
,.
continuity of
A
m
and
F
follows for any
;
OPIUM
b, v
XI -c<%)I<
-
t m(A).
F(A) >
and finally
If we eliminate the condition
of regularity, Vitali's theorem is no longer valid.
For the
system of all intervals this was shown in an example of
H. Bohr (B2) and another of S. Banach (Bl).
It also follows
from the work of Busemann and Feller (B3).
A number of proofs exist to show that nevertheless the
density theorem is valid with respect to the system of all
intervals and therefore that for any bounded function
DF(p) = f(p).
f(p),
The simplest of these is due to F. Riesz (RI).
Other proofs can be found in S. Saks (Sl).
The general problem of finding whether the density
theorem holds in Euclidean spaces with respect to an arbitrary family
R, has been treated by Busemann and Feller.
They assume that the family
R is such that to each point
p of the space tends a sequence of sets JPn of R (where
e
by tend we mean
=
.>lfor all
Cke)o).
For any measurable set
C)<$o1<j
with
n and the diameter of C
(Ca)
,
X with m(X)"; 0, and any 0,
letG,- g-(x)be the union of all sets f
I
?
<. R
> 0<
The following conditions are both necessary and suffi-
cient for the density theorem to hold with respect to the
system
R.
(1) //76
null sets.
zX*IV
(/)j*A
where
N1
and N2
are
Given a monotonically decreasing sequence of
bounded measurable sets
>
,
-
-
-? , ,
with
--
(
approaching zero, and a monotonically decreasing sequence
of positive numbers S, > it
dI//
- - >
0->
.
g -- with
-*0- 4/
The following condition is always sufficient for the
density theorem to hold, and is necessary and sufficient if
the family
R contains with each
all sets that are
similar and similarily situated to
(3)
and R
''7"
There exists a functionqe2 depending upon <
only, such that for every set
C6';w_)
fC/l
'O'
X with
m(X)
0
(X/
Busemann and Feller have shown (and we shall show more
directly) that (3)
in
Rn.
is satisfied by the set of all intervals
However the set of all rectangles in
Rn (i.e. with
sides not necessarily parallel to the axis) violates (2);
thus the density theorem does not hold with respect to this
system.*
From the validity of the density theorem with
respect to the set of all intervals in
Rn
we deduce this
important fact:
*See also N. Nikodym (Nl) who proves this fact in a different way.
-..-
,W..IIMILE.Hanomolliill
.L
.I 11
III111111
-Im.
.II. --,.
Given the function
Let
4
=
hXk
--
F(x,y)
=
(gqd
(
d
AK 2
in the differential quotient
C§E(Ai}>
)
P(,
F~x4
y Af-r
4))
=~
< C
ye'
/u(
)/4
YRxy#,(/ +i'F(Ayj
xe4yj
The existence of this limit followed from the fact that the
set of rectangles with I hk)< M
is regular.
Since the
density theorem holds for intervals we can replace
by
lim h lim k, i.e.
(
/,),,
W~K, i)) -
<f~orf~ Ay~frj
and similarly for higher dimensions.
6o 4+~ o
Jim (hk)
£
f,+
-
d
*For f(f q)unbounded wb cannot conclude what follwws from the
validity of the density theorem with respect to the set of
intervals, but a stronger criterion must be satisfied. . The
details are in Busemann and Feller. See also B. Jessen, J.
Marcinkiewicz, and A. Zygmund (Jl).
3y
A
F(-#4p)J
II. THE DENSITY THEDREM IN ABSTRACT MEASURE SPACES
In this chapter an attempt is made to generalize the
results obtained for Euclidean spaces to abstract measure
spaces.
Consider an abstract set
points
p.
1
Let
E
whose elements are the
be a Borel Field of sets of
as well as each point of
E
contains the space
E, which
Let
E.
be a completely additive, non.-negative set function,
m(A)
defined for the sets
A
.
of
are
The sets of
called measurable sets.
Assume that we are given in
point
a family &)
p
p.
the point
The
E
corresponding to each
of measurable sets
k'i/each
containing
(g/ will be called elementary sets.
(/
To eachl/w let there correspond a positive parameter
A sequence P/(,)of elementary sets is said to converge to
(a)
p, whenever
lim/Z 0/,*
0.
If the setSw9 contains
converging to
at least one sequence /kw9
system of elementary sets for the point
of a set
p
point
sets J
A
o we call -S99
p.
a
If to each
corresponds a system of elementary
, the totality of all S.f/ for all points of A
is called a system of elementary sets for the set
denoted by
A
and
S(A).
A subset
SI(E)C S(E) is itself a system of elementary
sets for the space
E, provided it still contains for each
point
now
p
of
E, a sequence of
J19;)
A be a measurable set and
the average density of
,~
='
A
p
converging to
a point of
A. We define
in the elementary set
C Anp
p. Let
VW9
v)
We define the upper and lower densities of the set
a fixed point
sets
S(E)
,AV
0
A
at
p, with respect to the system of elementary
as:
respecti4ey
=
as:
C
If
D =1
for almost all
for almost all
pFCA, we say that the density
S(E)
theorem holds with respect to the system
speak of the density
p F A, or
and we
D = D = D.
The following conditions have been established as
necessary and sufficient for the density theorem to hold
in an abstract measure space, with respect to the system
S(E) of elementary sets.*
Given a number
a, 0<ac1, a set
A
widai*a
and an arbitrary sub-system of elementary sets
A. Ther exists always a point
corresponding to
p
p
and contained in
(A) > 0
S'(E)C S(E),
A
and a set
S'(E)
such that
of
V/()
B. There exists always an enumerable sequence of elem*See R. de Possel (Pl). Condition B was established independently under a slightly different form by B. Younovitch,
(Yl).
C
I
p
entary sets
(/O)CS(E
corresponding to points
pn
of
A such that:
Letting
a = 1 -ii , condition A can be expressed as:
A'. Any measurable set of positive measure contains
an elementary set of S'(E) within S
measure.
Condition B can be shown to be a consequence of A. The
validity of the density theorem follows from B by replacing
the
l4".) by
disjoint sets VL- ) and deriving from this
an equivalent of Lebesgue's theorem for abstract spaces.
The proof of the density theorem then proceeds as before
(see Chapter I).
Given an arbitrar.
any
measurable set
I, m(X)
0, for
a., 0< a <1, denote by 6w- ,4K)the union of all elem-
entary sets ViC S(X)with
(a)
(b)
K/
)->0
'C"-
/
Observe that:
and if
(2)
Theorem I(a):
>
,,X,
(kr)
)
A necessary and sufficient condition for the
density theorem to hold with respect to the system S(E) of
**These conditions are identical with those of Busemann and
Feller for Euclidean spades except for the addition of
requirement (C) in part (b) of the theorem. The proofs
are also the same with the exception of the step in the
sufficiency proof of (2') which was condition (C).
E
elementary sets, in an abstract space
measurable set
X, with
is that: for any
m(X)- 0,
I
o
A neceszary condition for the density theorem to hold with
respect to the system S(E)
is that, given an arbitrary
sequence of measurable sets
X
- -
-
,-
X,
>
and an arbitrary sequence of positive numbers .
(2')
lim4i (G
--
p
- --
> E,>
-- E
L
CWoi
Proof:
lTe'
'Condition (1'): Necessity: for fixed a,
W contains
J>O
all elementary sets of arbitrary small parameter, of the
space
E
holds for
in which
D & a. TlIWa if the density theorem
S(E), (1') follows.
7F,
Sufficiency: if (1') is true,
OC.
x<
I
J>*
( +Nj,; m( 3) =4A/,)= and the density theorem holds.
Condition (2'): Necessity: for/>
CX
n:??(c'
but the density theorem gives:
and therefore
(K,)'since
lim C
= xn
lim'ICGd (AjPL 0.
Assume now that the system S(E)
of elementary sets
satisfies the following condition:
(C) Given an arbitrary sub-system
always possible for any measurable set
t o construct a sequence of sets
of the form:
S'(E)C S(E), it is
X, with
m(X>- 0,
Bn, each containing
X,
(x)#+q3
(b)
S'(E)
for all k and all a,
co0
such that7T Bn = X + N where N is a set of measure zero,
--*I =/
i.e. fi
(X) = lim m(Bn).
Theorem I(b): under the assumption (C) for the system S(E)
(2') is sufficient for the density theorem to hold with
respect to the system S(E) of elementary sets, and,
(3') if there exists a function Aik) depending only on
a (O<a<l) and the system S(E), such that for any measurable set
(, '
C'6
;;
X,
(
<
())
the density theorem holds with respect to
4)&v
S(E).
Proof:
Condition (2') Sufficiency: letA
set.
Given
a, 04 a<l, let A CA be the set of points in
which the lower density of A
to each point
(a)
-27
be a bounded measurable
p of
(Q)/,
Ctl
Denote for the points
(1 - a); i.e.
is less than
converges a sequence of Vft)with:
7 4/--
2
';: -
p of
j
I /}
the system of the /
Oj C ,<
satisfying (a) by S'(A ). By condition (C) we can construct
from the elementary sets of
S'( ) a sequence:
/=
(where we let:1Zd") <-I
with
7
and 2-0(k,)
en(V')
,,-<r,
for all
a, such that
+O;
0, let X ,,-)
-
o .
.')
Now since
+~G7K
m(A) zE
then,2x
C
'6-
,,-
(,
---)X,,.>.,
r
combining this with (a)
since
,)
is contained in the union of sets
A ,1 cn()j
)
satisfying
(CA)but:h7(Q~, (K-,))yand therefore,
0'0
0.
=
Condition (3'): Sufficiency: if (3') holds so does (2'),
V* now show:
thus the density theorem holds.
Theorem II; Conditions (C) and (2') (or (3') ) combined
imply
A
p(7 ).
X, with m(X)'>0, there
Proof: Assume that given a set
exists an
such that for all elementary sets
S(E)
where
Bn
V
of
S'(X)
(VA)
en (W
satisfies (C),
we can construct a sequence
(a)
Since
S'(E)cs(E)
a, and a system of elementary sets
CS
2with
z
say,
and
Consider the sequence:
25
we have A, :>
Now since
xC
thus: (b)
V
(d4&,
we have
del
L
-n
C
,
=
19
Bn
but
is the union of V, satisfying (b),
= 0.
lim(
Therefore
B
thus:
C 61,
lim m(Bn) = 0, which
contradicts our hypothesis.
Since
A
implies
B, and
B
implies
C, Theorem II
establishes the complete connection between the results of
de Possel and the results of Busemann and Feller when generalized to abstract spaces.
Remark: Condition
is always fulfilled if the space
C
is a compact metric space and the V
with the parameter
V
S'(E)<:ZS(E), each point
(
sets
S'(E)
S(E)
defined as the diameter
/Z
the elementary set
of
E
are open
4()
of
For, given any sub-system
.
p
of
E
is contained in a
) open, the system of elementary
,
forms an open covering of
E. Since
E
is
separable this covering contains an enumerable covering.
Thus given a measurable set
X, with m(X)>0, and a
sequence f, --- 0, we can cover
X with an enumerable
,, such that
number of sets
B17,
Condition B is always fulfilled if for any sub-system
St(E)e S(E)
the elementary sets of
for the Borel field $of
measurable set with
S'(E)
For, given a
measurable sets.
m(X)>0, we have ow
for all coverings of the set
X
form a basis
(Par
by elementary sets
Q(
C.S(E).
Given a sequence
m(X) - m(Bn))<
-> 0, let
E, ,
Bn =2i,
Bn-> X,
with
Bn.
this gives the desired sequence
Finally we state:*
Theorem II: a necessary and sufficient condition for the
DF(P)
derivative
Jf(P)
dP
of the indefinite integral
of any bounded measurable function
exist with respect to the system
and to equal
f(P)
S(E)
F(P) =
f(P)
to
of elementary sets
is that the density theorem holds with
respect to this system.
*See proof in R. de Possel
~I
III. THE DENSITY THEOREM IN FINITE PRODUCT MEASURE SPACES
El,
Given a finite number of abstract measure spaces
E 2, . . . En: let
the Borel field
70
=TEi
E
be their product.
spanned by the sets
A
in
Consider
E
which
are the direct product of measurable sets in the component
spaces.
(A
=TAi,
Ai V.Ei,
Ai measurable.)
It is
possible to define a completely additive, non-negative
measure for the sets of .
such that
m(A) =27Fm(Ai).
The proof can be found in S. Saks (Sl') -or a more elementary proof in Lomnicki and Ulam (Li).
Theorem I:
Let
I
and Y be two abstract measure spaces
S1 , S2
in each of which ther is defined a family
subsets
l.
Given a number
arbitrary measurable sets
existence of functions
(yk
where(,
such that
6~
X
Z
o9
Y. Define the family
arbitrary measurable in
ein))<
)being
C S2
.. >
(
as the collection of all sets{5 =
We assert: for
'/
such that
o( at,,
Consider the product space
R = Si X S2
(a),
assume the
the union of all sets{C Sl,
(.>
17
a, O< a< 1, and
X' C X, Y' C Y,
9(), (
of
X X Y,
r
the union of all sets
X
0l
o < /hZi
<<
6i
for which Ai-7
2/
(C-l
Proof:
Let a
be a set&64)
Q
For any set
x C
x
.
Let
Xo
>
Rr
>& - -
<
(a/2-7(Ko)
2=
ct')A'e
for:
=: Z
-(x)
(ek
6
Now let§
We have for any
Y
and therefore
C
i.e.
of
for which
X)
O' r<a. We prove -
C
9A(A'
(e) for all
, (ex) =
2
5
'C
L'
let:
be the set of all
0" Ar6v
where
X
We can write
We have
i.e.
_
C27
C
G
1
E
but
X
(A
giving the desired result:
Similarly for the product of n
with functions
'
c)
-
-
spaces
4
(2)) <"r
'
S1, S2
C
7, /h--
S2
Let the families of subsets
x
-
,4
be two given systems
of elementary sets in
X
and
Y.
6< a <ae
Assume that condition (C) of Chapter II is satisfied for the
systems
s(X),
S(Y),
S(X) XS(Y).
and
asserts that if in both
X
and
Y
Theorem I then
inequalities of the
,
)
- -
form (3') hold and therefore the density theorem holds with
respect to
S(X)
and
S(Y),
then the density theorem holds
with respect to the system of elementary sets
in
..
XX Y.
Let now X
x
on the line;
points in
A
S(X)? S)
x1 , x2 ,
be a finite set of points
let
m(A), AC I
be the number of
divided by the total number of points in
i.e. the probability of finding at random a point in
With each point
xi
associated all sub-sets
Yi
A.
x
of
and which are such that with two points
xi
and
they contain all intermediate points (i.e. all
xr with
sets
V-
k.< r<j).
Let
S(X)
for all points of
Take
A
arbitrary in
ding each point
xi
of
points to each side of
A
xi,
con-
xj
taining
xk
X,
be the totality of all such
X.
X. C 'l)is obtained by inclu-.
V
in a
extending
1/a
(3
i.e. of total length
This result is also true if X
is infinite and
is replaced by the appropriate measure defined on
S(R1)
-)
m(A)
We
X.
being the
now apply this result to the space
R1,
family of all open intervals
on the line.
Replacing
points by intervals in the preceding reasoning, we have
for an arbitrary measurable set
X: Pi
(6$
<
__
-,
combined with the first part of the remark in Chapter II,
this gives:
*Note that S(X)x S(Y) determines a system of elementary sets
in the space Xx Y, if convergence in X X Y is defined in
terms of convergence in the factor spaces X and Y.
Theorem II A: the density theorem holds in
of all intervals.
Rix R1 = R2 :
ZcRn
for the system
Applying Theorem I to the product space
s(R2 )
is the set of all
intervals in the plane.
For
R-
For any
Z C R2
X
,
i.e. all
and measurable,
and measurable
i.e. Theorem II B. The density theorem holds in
2.
Rn.
z
IV
THE DENSITY THEOREH IN INFINITE MEASURE PRODUCT SPACES
Let
E
E =
be the direct product
number of abstract measure spaces
Consider the Borel fiel
A
Ei
of an infinite
of total measure 2.
Ei
panned by the sets
A
in
E
which
are cylinders over a finite number of dimensions (i.e. sets
of the form
TE;,
S =Tix
Ameasurable.
It is
possible to define a completely additive, non-negative
measure for the sets of
-
such that
ti(A)
The
proof of this fact has been established under various restrictions by the following people (among others):
Steinhaus(S3):
Ej
is the unit interval (0,l).
The equiv-
alence of the measure in the infinite dimensional space to
the Lebesgue measure on the line is shown by extending to an
infinite number of dimensions the usual 1:1 mapping (within
a null set) of the points inside the
n
dimensional unit
cube onto the unit interval.*
Kolmogoroff (K2):
space
Ei
Use is made of the fact that each factor
is compact and that consequently E
compact if convergence in
E
is also
is defined in terms of conver-
gence in each component space.
Doob (Dl): Ej is any abstract measure space, but this case
*That is, to
,
,
,Cwhere j9 0o, 1,,
Z-6,-f,
corresponds the point
o1
0 19,,, 9 t9
on the
line, and vice versa.
11
3i,&
9,3- on the
oaI
is reduced by a set isomorphism to that treated by Kolmo.
goroff.*
Kakutani (Kl):
The
Ei
are any abstract measure spaces.
The proof given by this author is the most straightforward
of all and does not use any concepts of topology.
Jessen (Jl):
The
are the unit circles,
Ej
E
is called
De la Vallee Poussin's
the infinite ,dimensional torus space.
notion of dissections and nets is extended to this space in
the following fashion.
D,
A dissection
is defined by
removing a finite number of points from each of a finite
number of circles
E1 ....B,
and considering all the inter-
vals that can be formed by taking direct products of arcs
in the
Ei, so obtained, by
TEi.
A subdissection
D2
removes additional points and contains all possible resulting intervals, including those of
dissections
D1 , D2,.*.*..Dn*** such that for any fixed
the maximal length of each arc of
Dn ->o0
as
A sequence of sub-.
Dl.
n -- '0o
Ek
k,
which corresponds to
generates a net in
E.
With the aid of so-called corresponding nets in
E
and the unit circle** there exists a 1:1 mapping (within a
*The complete additivity of the measure in E .(i.e. A = (Ai)
m(Ai) )'is
A rE, Ai 1lAJ = 0, i $ J to show m(A) =
proven by performing a measure preserving mapping in which
the enumerable number of sets Ai involved become determined each by a set of inequalities imposed upon a real
valued function. To the Ai thus correspond an enumerable
num1W of disjoint sets A of equal measure in the space
E =TT Ei whose factors are the unit intervals Ei. See
L
null set) of
El, which is measure preserving
back onto
E
E, into meshes of the corres-
and maps meshes of the net in
ponding net in
El.
From the existence of this mapping is concluded:
E
(a) The equivalence of the measures in the torus space
and the circle
El.
(b) Given a net, denote byA
Q(for any value of
stepfunction which in any interval
In
n, the
n-th dissec-
of the
=
is equal to the corresponding quotient
tion
Dn
with
F(A) =
f(x)dx
AC E
then
almost everywhere.
limL
"
,(x)
=
f(x)
0O
We are however not so much interested in knowing
whether the density theorem holds with respect to a particular family of intervals in
E, such as a net, but
rather whether it holds with respect to Wh(family of all
intervals in
interval in
E, i.e. all sets of the form
Eij, where only a finite number of the
different from the whole space
by
TAi, Aj
x = (xlx2****xn*...);
E1.
Ai
an
are
Denote a point of E
Definition: a sequence of
points in E, x, x 2 , .... xn,.... converges to x, if and
Halmos and von Neumann (Hl) for a clearer exposition of this
mapping.
dn
"That is, such that for each In c Dn, there is an in
(dissection in the unit circle) and vice-versa with
m(In) = m(in) and In+l C In implies in+IC- in'
**MSteinhaus' method of proof is merely a particlar case of
this one.
only if
the corresponding sequence of coordinates
x
xi,
i.
for all
We can, but do not need here, give an explicit expresE, in terms of distance in each of its
sion for distance in
we merely assume that the diameter of an
component spaces;*
interval in
E
tends to zero if and only if the diameter of
El, El;X
yE
each of its projections on the spaces
2
,**..-etc.
0.
-+E
Lemma:
,
can be written asX
If X C E
h
contains an interval of arbitrarily small
EZ, and
diameter, then
xCv,X-
+~()
G)-.-
V
4
and
-IV
k
for all
X
-
where
V =
for let
-
then
)
/,(<
It is easy to see that:
Theorem I: Condition (3)
product space E
=1
is never fulfilled in the infinite
E(o,).
For consider a set of the form
where the
Xi
follows that
,
X =
k
-
are intervals in
Ei; from the lemma it
M 6(
(C
C A))
*See for instance Frechet (Fl).
--- )
'
(,-)-
~
-
thus
limdo#
,--;7
-
lim
:
The same is true if
=V.
i
E
is the
07 -;14h
infinite product of an abstract measure space by itself in
which the function-A)of theorem I is ,> 1. Elementary sets
in E
are defined as sets of the form
elementary in
Ei.
c
h
X2
.
=..-X.=
for any set
/
LetI(vX)>
X1 =
X
,rK.
X =
Now take
AC E
be given by
:(
'wher
.
2
and
>
Xn, then.
e/, it is always possible to
again, given any function
find a set
77
V
withn
(/2) > ffCW))/.
Condition (21) is also not fulfilled in the space
E:
the following is an example of a sequence of measureable
--
'
sets:
,)-o,-and a sequence
rX
S>A
S4,j;&z-
IS712r
-
>.
>
-with
07
for all
k
and
+ 0
lim
where
is the unioh of all intervals
in
E
with
0<x<a
x0
;gx,
X, =
s
63
Jr
C44)
IX
2XE
42.£I~
'an r
(dand
Denote the interval
+f C,
such that:
~
by
(0,a).
Let
4
s
n
5
K
E
Z
I~
x
x
-
4(>~a
)~
;I
K(,L)(o±)
I -(
wl10';,,,
dv
-;p
K
4k
(-4L/
,z
3
-~~g
CXe 2 . k1e
fi7 A
.
x
~k~
/
(rc-1 ,
3
Are"S "'-o
37
-7'.
ey
It
x~Ar'6
(aI)JoIx
k
q
-.
IJ~-~-4AXL~KO,
+C~k
+Ex~z. x~se
(oL,01J)[
-C2
4qx4r Es
x
(
.. /
x
S
X
3 X§
xX
Ex3 k ' E
a
x
(
,x(
X8
X r4O
9
J-4
)
X
(0,
K
sz
.i)>g$
)C4
)x(
-L
x("! g$Oj)
.,x
-
,C-k~
-xQ
V
/
,x
)
x(",
Ef--
'V({()
Z
C'
E -t.
(o
Er
ecja
r,
--
E
p
etc.
This construction can be explained as follows:
consists
(2n)2n = p(2 11) terms, each term being the
of the sum of
product of
Xn
factors equal to the interval
2n = k(2n)
by the remaining spaces in their
entirety.
In generalA'
is obtained from
fashion: with each term ol
terms of
first 2
.
A
X,
in the followi ng
2..
we associate
By "associate" we mean that we take the
factors of each term of
contained inside
74'r
I
the
2n
X
factors of the term of
with which it is asso-
"ii4
ciated; the other 2
being taken each time inside an
"utilized" before.
(2-
we utilize
El
and
4
8
X
g 2#2 7,6 in the fourth term.
= 64 terms associated with each term of
£
and utilizes in addition
up to,
which has not been
E3
Thus in forming X
E2 in the2.- first term, up to
has
, different from (0,l)
factors of
6
up to
A
.
-Xutilizes
, etc.
We have for this sequence:,,
(S
(S,,))
whereY(X,)is the sum of all overlapping parts of the
terms of
; i.e. -, 6(KXis the measure of the sets that
are added up more than once in obtaining the first
term:
.
2.
A somewhat lengthy calculation** shows that m
is of a higher order in
thus
limm(X)=
n
than(
(S(x4)
VNI (m ( X
= 0.
N1 -->to
Now using the lemma on page 2.1 and (4) on page &
we have:
0(j
-j)
0C>C 0
-f -
-
C
E
X Ey --
Cetc.
**See appendix, where we shall also explain how the above
construction was arrived at.
(X)
and in general] G
contains for each term the sum of the
products of< of each factor by all the other factors.
(See
Now certainly:
P-- 0
is the set of all parts of gc
where
X )that are
added up more than once due to overlapping of the terms of
O(
-.
.
2 h
(SC
It is easily seen* that
Finally take~>-
5)
where
- ---
The sequence
-
0 for
lim
is such that
i;
all
Taking now for the set
)
X J1
for a term of the form, say:
,where--
K; t
~'A
4
~i~i~~
~
e+c.
the sum of a sufficient number of elementary sets of the
form
e-3
CX
C~
~x
-fi.
.-
Cwx-.
('
c'
for all
with cJ())
so that their sum equals e; ()(
another set of
with first:
xX
will sum
)
i,
then
(?( X X'"k---
etc.
xSee appendix.
**Note that by the lemma of page 21 this is all that is
required of the components of C
in C, (X.
for
e
to be contained
-~
We find that in
and
lim
i1
;(6
Theorem II:
()
060)
A.E.D.
E =
In the space
are the unit interval
(0,l)
Ei, whose factors
Ei
the density theorem does not
hold with respect to the set of all intervals in
E.
The definition of a Lebesgue integral in a space of an
infinite number of dimensions has been given by Daniell (Dl)
or by Jessen among others.
S~~
-,
of
Consider such an integral
f
{t
~1 .
X(.
a
we".. - -
'.,
--,-t,; -- d4 61-,L and a
7
=
sequence of intervals
), whose diameter tends to zero as n -0:
(where/ I,,V
It is not generally true that
Theorem III:
lim ..
-
X
Ap
For otherwise letting
function of a set
true in
If
E
, r
- -. ,,--)
be the characteristic
A, the density theorem would always be
for the set of all intervals.
E =
Ei
with the
Ei
finite abstract measure
spaces which are such that:
(a) Sets of any measure between zero and one exist in
each of the
Ei;
(b) The functionffi,
defined byn,
(ejI)) =f4 (
Xfor a measurable set
X
in
j40is greater than one;
Ei,
'
then we can construct an example similar to the one above to
show that the density theorem does not hold with respect to
the family of all elementary sets in
E.
In general we must know something more about the elementary sets (i.e., about-qoc}) and the measure space itself
before the analogue of Theorem II can be established.
Theorem IV:
E =TFE±
There exists a measurable set
A
in -the space
(Ei the unit interval) and a family St(E) of
intervals in
E, containing for each point
sequence of intervals converging to
p
of
p, such that
A
a
S (A)
Vn,
contains no enumerable collection of elementary sets
covering
A,
with) /r
A)-
4P" ( V
for E
positive
and arbitrarily small.
This follows from condition B of R. de Possel (p.
7 ).
--JI
APPENDIX
I. Proof that
m S()
=O(
As a first, and too large approximation of m(S(Xn))
we
find the sum of the measures of the parts in common to the
terms of
when combined two by two.
Xn
The manner in which the measure of the part common to
Xn
two terms of
is calculated is most easily illustrated
by an example:
(14x
A =
Say
x(
- - - x'
(
44&Y 'Y
B=(
then
AB =
L}
m(AB) =(o
i.e. it
is the product of those factors different from
(0,l)
common to the two sets by the remaining factors different
from
(0,1)
of each of the two sets.
Now, in the set
(1)LI2")
Xn
we have:
in common and there are in
T(2
(2)
factors
terms which have two by two
Xn
the sum of
collections of such terms.
groups of
.?
K p2
^
)
terms
such that each term from one of these groups has .I'
tors in common with each term of another of the groups.
fac-
1
12i)such
There are
(3) r(2'')
collections of groups.
terms such that each term
groups of
of one of these groups has 27
factors in common with each
There are PcL.'~) such collections of
term of another group.
groups.
*
(n)
*
0
0
pr.'%)
*
4P
*0
00
0
groups of
*
0
0
0
0
0
0
0
0
0*
(tt)terms such that each of one of
these groups has no factors in common with any term of the
other group.
We now recall the formula:
Given
k
groups of
n
objects each, the total number of
different combinations of objects, not belonging to the
same group, is given by:
-_)
__
We thus get:
1
m(S(Xn))
__2_
2--
PC 2.O
4
.2.ell%
)
2_tt~
[
kZO
A,
a"
Ic
Le ---
-ti
3
IV
ChapVe
so that m(X
)c
each term of
th
diec
while
LbA)
h
is replaced by
Xn
prdutofk
( 2 11)
(A2 -
esA
CXh
for
-
terms giving:
ckwz
h
II. Procedure Used in the Construction of the Example of
Chapter IV
We know by Theorem I, p.2-1 that for a set A which is
the direct product of
C^ (a C
k sets,
Al XA2
To utilize this fact for the
L) , A)
construction of a sequence of sets ?(,
m(Xn) -* 0
with
particular if
X
we would have:
the latter expression
more slowly than
- - x
m(Xn).
In
xE
"X
X
lim A.(
If the factors
demand
-
),, since
*CC'
some of the
n
'-x,-
we had:
X.%X
and
LX:)>0,
lim m(6'
must tend to zero with
XAk3 we have
.
of
Qare
)is
; e
-->a
at least
for all
for some
intervals, the diameter of
i
i
while
.
m(Xn) =O(-)
at least, and thus t(X
But we
implies
If the factors
of Xn
are the sum of intervals whose
length tends to zero( in which case we can satisfy
C
whose sum is
,and
form
,
satisfy
-
Y
,
'
((
,)-e
say, intervals of the
it can be shown to be impossible to
')*-
.
Xn
As a next step we then take
as a sum of
oo,)dis-
joint terms each of -ko') factors whose product equals
where we require
(3)
The measure
of Xn
~f
tends to zero:
Now by (2)
each factor of each term
1/
C.
-c
this can be achieved by taking ko%) = n,
L.- ) z
in Chapter IV.
1 cmt)
= nn.
Thus
which leads us to the example we displayed
(We use
instead of nn.)
BIBLIOGRAPHY
(Bl):
Banach, S. : Sur le theoreme de M. Vitali, Fundamenta
Mathematicae i
p. 130 (1924)
(B2):
Bohr, H.: see Caratheodory, Vorlesungen uber Reelle
Funktionen, Druck and Teubner, p. 689 (1927)
(B3):
Busemann, H. and Feller, W.: Differentiation der
Lebesgue Integrale, Fundamenta Mathematicae 12
p. 226
(Cl):
(1934)
Caratheodory, C.: Vorlesungen uber Reelle Funktionen
p. 492
(1927)
(Dl):
Daniell, P. J. : Integrals in an infinite number of
dimensions, Ann. of Math. (2) 20 p. 281 (1919)
(D2):
Doob, J. L.: Stochastric processes with an integral
valued parameter. Trans. Amer. Math. Soc. 44 p. 87
(1938)
(D3):
Dunford, N. and Tamarkin, J.: A principle of Jessen
and general Fubini theorems. Duke Math. J. 8 p. 743
(1941)
(Fl):
Frechet, M.: Les Espaces abstraits, Gauthier Villars,
p. 164
(1928)
Sur quelque points du Calcul Fonctionnel,
Rend. del Cire. Math. di Palerno, 22 p. 1 (1906)
(Hl):
Halmos, P. and von Neumann, J.: Operator methods in
p. 332
classical mechanics II, Ann. of Math. 4
(1941)
(Jl):
Jessen, B.: Theory of integration in an infinite number of dimensions, Acta Math. _1 p. 249 (1934)
(J2):
Jessen, B., Marcinkiewicz, J. and Zygmund, A., Fundam.
Math. 21
p. 217 (1935)
(Kl):
Kakutani, S: Notes on infinite product measure spaces I
Proc. Imp. Acad, Tokyo _9 p. 143 (1943)
(K2):
Kolmoporoff, A.: Grundbegriffe der Wahrscheinlichkeits
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mesure dans les espaces combinatoires et son
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d'ensemble. J.de Math. pures et appliquees, 9ieme
s, 12 p 391 (1936)
(Ri): Riesz, F.: Sur les points de densite au sens fort,
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Theory de l'integraleproof of the densiw
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$2):
Sierpinsky ,W: Demonstwation elementaire du theoreme sur la densite des ensembles, Fund.Math.4
£. p.167 (1923)
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BIOGRAPHICAL NOTE
The author of this thesis was born on August 28, 1924
in Scheveningen, Holland.
She attended the Montessori
school in the Hague. Her secondary education was received
at the College Marie Jose in Antwerp, Belgium and at the
Lycee Francais de New York (from 1940-.41) where she passed
her baccalaureat examination.
From 1941 to 1943 she attended
New York University and obtained a B.A. degree.
She entered
M. I. T. as a graduate student and research assistant in
physics in November, 1943, and obtained a M. S. degree in
Physics in February, 1945.