Part V
Applications
16 Loeb Measure
Measure theory studies operations that assign magnitudes to sets, like mea
suring the length of an interval, the area of a plane region, or the volume of
a solid; counting the number of elements in a set; calculating the probabil
ity of an event in a sample space or the definite integral of some function
over a set; etc.
Now, the "measure spaces" on which such operations are defined are
typically closed under countable set unions, and this feature is fundamental
to the theory. But an internal collection of sets typically fails to be closed in
this way. However, in 1973 Peter Loeb discovered that this very failure could
be exploited to give a new way of constructing standard measure spaces
out of nonstandard entities. 1 This has led to some interesting applications,
particularly in probability theory and stochastic analysis. For instance, it
provides a representation of Brownian motion as a "random walk with
infinitesimal steps"n.
We will now develop Loeb's construction, elucidating the role played in
it by the nonstandard principles of countable saturation, sequential comprehensiveness,
and overflow. We will then apply it to show that Lebesgue
measure on the real line can be represented by a weighted counting measure
on hyperfinite sets, using infinitesimal weights.
But first, a review of some of the basic concepts of measure theory.
1See example 6 of Section 16.1 and example 3 of Section 16.3.
204 16. Loeb Measure
16.1 Rings and Algebras
A ring of sets is a nonempty collection A of subsets of a set S that is closed
under set differences and unions:
• If A, B E A then A -B, A U B E A.
It follows that 0 E A, since A-A = 0, and that A is closed under symmetric
differences AB and intersections A n B, since
AB (A-B) U (B(-A), and
AnB A-(A-B).
An algebra is a ring A that has S EA and hence (indeed equivalently) is
closed under complements Ac =S-A. If A is a ring, then AU {S(-A:
A E A} is an algebra, the smallest one including A.
A a-ring is a ring that is closed under countable unions:
• If An E A for all n EN, then UnENAn EA.
The equation
nneNAn = A1 -(UneN(Al -An))
shows that a a-ring is also closed under countable intersections.
A a-algebra is a a-ring that is an algebra. The intersection of any family
of a-algebras is a a-algebra. Thus for any A P(S) there is a smallest
a-algebra S(A) that includes A. This S(A) is the a-algebra generated by
A.
Here are some examples of these concepts:
(1) P(S) itself is a a-algebra.
(2) If S is infinite, then
• the collection of all finite subsets of S is a ring that is not an
algebra;
• the collection of all finite or cofinite subsets of S is an algebra
that is not a a-algebra;
• the collection of all countable subsets of Sis a a-ring that is not
an algebra when S is uncountable.
(3) Let CIR be the collection of all subsets of lR that are finite unions of
left-open intervals (a, b] = {x E lR : a < x :::; b} with a, b E lR and
ao:::; b. (Thus 0 = (a, a] E CJR.) CIR is ring in which each member is in
fact a disjoint union of left-open intervals (a, b]. CIR is not an algebra,
and is not closed under countable unions: (0, 1) is not in CIR, since each
member of CIR will have a greatest element, but (0, 1) is the union of
the intervals (0, 1-] for n EN.
16.1 Rings and Algebras 205
CIR does, however, contain certain significant countable unions: for
instance (0, 1] is the union of the pairwise disjoint intervals ].Any reasonable notion of measure should thus assign to (0, 1] the
infinite sum of the measures of the intervals ].
(4) Let BR be the O"-algebra generated by CR. Each open interval (a, b) in
.IRis in BJR, being the union of the countably many left-open intervals
(a, b-] for n EN. Hence every open subset of lR is in BR, being the
union of countably many open intervals (take ones with rational end
points).
On the other hand, if a O"-algebra contains all open intervals, it must
contain any left-open (a, b] as the intersection of all (a, b + ) for
n EN. Thus BJR is also the O"-algebra generated by the open intervals,
as well as the O"-algebra generated by the open sets of JR.
The members of BJR are called the Borel sets.
(5) Let S = {1, ... , N} with N an unlimited hypernatural. Then S is
hyperfinite, and the collection Pr(S) of all internal subsets of S is
an algebra (also hyperfinite) that by transfer of the finite case will
be closed under hyperfinite unions, i.e., unions of internal sequences
(An : n ::; K) for K E *N. Pr(S) is not, however, a O"-algebra: it
contains each initial segment {1, ... , n} with n E N, but does not
contain their union because that is the external set N.
This same analysis applies to the algebra of internal subsets of any
nonstandard hyperfinite set S = {sn : n::; N}.
(6) Let A be an algebra in some universe liJ. In any enlargement of lU,
*A will be an algebra, by transfer, but in a countably saturated enlargement
*A will not in general be a O"-algebra, even if A is. To see
this, let (An : n E N) be a sequence of members of *A with union A.
Each An is internal, and if A were in A, it would also be internal and
hence by countable saturation would be equal to Un k, Amn= 0, since
Un:::;kAn and Am are disjoint, and so J.tL(Am) = 0. Hence
UnENAn A1 U· · · U Ak, and
LnEN J.tL(An) J.tL(AI) +n· · · + J.tL(Ak),
from which it follows that Jl.L satisfies Ml.
(4) Let A be an internal ring of subsets of some internal set S in a
countably saturated enlargement, and let J.tn: A *[0, oo] be a finitely
additive function. Adapting the construction of (3), put
{ sh(J.t(A)), if J.t(A) is-limited,
Jl.L(A) =
oo, if J.t(A) is unlimited or oo.
Then reasoning as in (3), we show that Jl.L :A [0, oo] is countably
additive, and so is a measure on the ring A.
(5) This last construction has (3) as a special case, and also covers other
natural extensions of (3) that involve hyperfinite summation. Let w:
S *JR be an internal "weighting" function on a hyperfinite set S.
For each A EPI(S) put
(recall the definition of hyperfinite sums in Section 13.19). Then J.tw
is a "weighted counting function" that is finitely additive and induces
the measure J.t£ on PI(S).
In fact, every internal finitely additive function J.t : PI(S) *[0, oo]
arises in this way: put w(s) = J.t({s}). Example (3) itself is the special
case of a uniform weighting in which each point is assigned the same
weight w(s) =
16.3 Outer Measures
We now review the classical procedure of Caratheodory for extending a
measure J.t on a ring of sets A to a measure on a a-algebra including A.
If B is an arbitrary subset of the set S on which A is based, put
Here the infimum is taken over all sequences (An : n E N) of elements
of Athat cover B. The function J.t+ : P(S) [0, oo] is called the outer
measure defined by J.t (although it may not actually be a measure). It has
the following properties:
16.3 Outer Measures 209
• J.L+ agrees with J.L on A: if B E A, then J.L+ (B) = J.L( B).
In particular, J.L+(0) = 0.
• Monotonicity: if A B, then J.L+(A) J.L+(B).
• Countable subadditivity: for any sequence (An) of subsets of S,
• For any B S and any c E JR+ there is an increasing sequence
A1 A2 · · · of A-elements that covers Bnand has
A set B S is called J.L+ -measurable if it splits every set E C S J.L+additively,
in the sense that
For this to hold it is enough that
whenever J.L+(E) < oo.
The class A(J.L) of all J.L+ -measurable sets has the following properties.
• A(J.L) is a a-algebra.
• A A(J.L), i.e., all members of A are J.L+ -measurable. Hence A(J.L)
includes the a-algebra S(A) generated by A.
• All J.L +-null sets belong to A(J.L) .
• J.L+ is a measure on A(J.L), and hence is a measure on S(A).
• If J.L is a-finite on A, and A is an algebra, then J.L+ is the only extension
of J.L to a measure on S(A) or on A(J.L).
Because A(J.L) contains all J.L+ -null sets, J.L+ is a complete measure on it,
which means that
• if A B E A(J.L) and J.L+(B) = 0, then A E A(J.L).
This entails that
• if A, BE A(J.L) with A Bnand J.L+(A) = J.L+(B), then any subset of
B-A belongs to A(J.L) (and is J.L+-null), and hence any set C with
A C B belongs to A(J.L) and has J.L+(c) = J.L+(A) = J.L+(B).
210 16. Loeb Measure
16.4 Lebesgue Measure
Lebesgue measure is defined by the outer measure ..\ + constructed from
the measure A on CIR that is determined by putting ..\( (a, b]) = b -a. The
members of the a-algebra CJR(..\) are known as the Lebesgue measurable sets
and include all members of the a-algebra BIR of Borel sets generated by CJR.
We will write ..\(B) for ..\+(B) whenever B is Lebesgue measurable.
Some facts about Lebesgue measure that will be needed are:
(1) A is the only measure on BIR that has ..\((a, b))U= b-a: any measure
on an algebra including BIR that agrees with A on open intervals must
agree with ..\ on all Borel sets.
(2) For any Lebesgue measurable set B there exist Borel sets C, D with
C B D and ..\(Dn-C) = 0, hence ..\(B)n= ..\(C) =n..\(D).
(3) A set B IR is Lebesgue measurable if for each c E JR+ there is a
closed set Cc Band an open set De :2 B such that ..\(Den-Cc) ..(B)n= JLL(S n *B)
holds in general, but that suggestion is quickly dispelled by the case B = Q.
Every grid point is a hyperrational number, so S *Q and hence JLL(S n
*Q) = JLL(S) = oo, while >..(Q) = 0.
Rather than S n * B, the appropriate set to represent B in S is the set of
grid points that approximate members of B infinitely closely. This is the
set
{s E S : s is infinitely close to some r E B}
{s E S: sis limited and sh(s) E B},
which may be called the inverse shadow of B. The definition of sh -l(B)
uses a condition that is not internal, so the set itself cannot be guaranteed
to be internal, and more strongly may not be Loeb measurable, i.e., may
not belong to A(JLL). One case in which it is not internal but nonetheless
is Loeb measurable occurs when B = : since
sh-1(R) = {s E S: sis limited}n= UnEN(S
n *(-n,nn)),
while each set Sn*( -n, n) is an internal subset of Sand so belongs to A, it
follows that sh-1(R) belongs to A(JLL) by closure under countable unions.
But sh-\R) cannot be internal, because it is bounded in * but has no
least upper (or greatest lower) bound.
The general situation is this:
16.8 Lebesgue Measure via Loeb Measure 217
Theorem 16.8.2 A subset B of JR is Lebesgue measurable if and only if
sh-1
(B) is Loeb measurable. When this holds, the Lebesgue measure of B is
equal to the Loeb measure of the set of grid points infinitely close to points
of B:
>.(B) = J.LL
(sh-1
(B)).
Proof Let M = {B JR: sh-1(B) E A(J..t£)}. FornB EM, put
v(B) = J..lL
(sh-1(B)).
Our task is to show that M is the class CJR(>.) of Lebesgue measurable sets,
and that v is the Lebesgue measure >..
By properties of inverse images of functions,
sh-1 (0) -0,
sh-1(A-B) sh-1(A) -sh-1 (B),
sh-1(UnENAn) UnENsh-l(An)
·
Since A(J.LL
) contains 0 and is closed under set differences and countable
unions, these facts imply that M has the same closure properties. Since
sh-1n(1R) E A(J.LL
), as was shown above, we also have JR E M. Altogether
then M is a a-algebra, on which v proves to be a measure.
At this point we need the the following lemma.
Lemma 16.8.3 M includes the Borel algebra BJR, and v agrees with Lebesgue
measure on all Borel sets.
Proof Each open interval (a, b) JR belongs to M, since sh-l ( (a, b)) is
the union of the sequence (An : n EnN), where
A = S n *(a + .! b -.! ) E A.
n
. n' n
But BJR is the smallest a-algebra containing all open intervals (a, b), so this
implies that BR M. Also, by Theorem 16.8.1,
J.LL(An) = (b-)-(a+) = b-a-
'
and hence as the An
's form an increasing sequence,
v((a, b))o= j..tL
(sh-1
((a, b))o= lim j..tL(An) = b-a.
noo
Thus v is a measure on BJR that agrees with A on all open intervals. But
any such measure must agree with >.non all Borel sets (16.4(1)). D
Now, if B JR is Lebesgue measurable, then (16.4(2)) there are Borel
sets C, D with C B D and >.(C)o= >.(B)o= >.(D). Then
sh-1n(C) sh-1n(B) sh-1
(D),
218 16. Loeb Measure
and by Lemma 16.8.3, C,D EM, whence sh-1(C),sh-1(D) E A(JLL), and
JLL(sh-1(C)) = v(C) =n>.(C)n= >.(D) = v(D) = JLL(sh-1n(D)).
Since J.lL is a complete measure on A(JLL) (by the general theory of outer
measures), it follows that sh-1(B) E A(JLL), and hence BE M, with
1n
v(B) = JLL(sh-1(B)) = JLL(sh-(C)) = .A(C) =n>.(B).
This establishes that every Lebesgue measurable set is in M (i.e., CIR(.A)
M) and that v agrees with >. on all Lebesgue measurable sets.
It remains now to show that M CIR(.A), and for this we need the
result from Section 11.13 that the shadow of any internal subset of *IR is
topologically closed as a subset of IR, and so is a Borel set.
Let B E M, i.e., sh-1(B) E A(JLL)· First we consider the case that
sh-1(B) is JLL-finite and show that B is Lebesgue measurable by the criterion
16.4(3). But if JLL(sh-1(B)) < oo, then by Lemma 16.6.1, sh-1(B) is
JL-approximable, so for any given c E JR+ there exist sets C, D E A with
C sh-1(B) D and JLL(D)n-JLL(C) ..(B) of a subset B of lR can be obtained as the
Loeb measure JLL(sh-1(B)).
D