Introduction to nonstandard analysis and Loeb measure theory

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Introduction to nonstandard analysis and Loeb measure theory

Before deﬁning Loeb measures, we brieﬂy introduce the nonstandard universe in which we will be working. This exposition is largely based on the very clear monograph of Cutland [7]. Although Loeb measures are standard measures, their construction involves nonstandard analysis (NSA).

The hyperreals

We construct a real line ∗R which is richer than the standard reals R. This is an ordered ﬁeld which extends the real numbers in two notable ways:
(i) ∗R contains non-zero inﬁnitesimals; that is, numbers of which the absolute values are smaller than any real number; and
(ii) ∗R contains positive and negative inﬁnite numbers; that is, numbers which in absolute value are larger than any real number. We say that x,y ∈ ∗R are inﬁnitely close whenever x − y is inﬁnitesimal and denote it by x ≈ y. Thus, x ≈ y if for every ε > 0 in R, |x−y| < ε. The set of all such y that are inﬁnitesimally close to x is called the monad of x. There are several ways of constructing the extended universe. We shall use an ultrapower construction. An axiomatic approach is also possible, as developed by E. Nelson; see for instance [30]. We prefer to use the ultrapower construction because it is pertinent to later constructions. Deﬁnition 2.1. A free ultraﬁlter U on N is a collection of subsets of N that is closed under ﬁnite intersections and supersets, contains no ﬁnite sets and for every A ⊆ N has either A ∈U or N\A ∈U. Given such a free ultraﬁlter U on N we construct ∗R as an ultrapower of thereals ∗R = RN/U. The set ∗R that we obtain therefore consists of equivalence classes of sequences of reals under the equivalence relation ≡U, where (an) ≡U (bn) ⇔{n : an = bn}∈U. The equivalence class of a sequence (an) is denoted by either (an)U or, in the sequel, by haniU. It is clear that ∗R is then an extension of R, the usual real numbers represented by equivalence classes of constant sequences. The usual algebraic operations such as +,×,< are easily extended, but shall be denoted in the usual way. Functions and relations on R can be extended pointwise without diﬃculty. Exactly which properties of ∗R are inherited from R is speciﬁed in the following theorem, a restricted version of the more general transfer principle: Theorem 2.1. Let ϕ be any ﬁrst order statement. Then ϕ holds in R if and only if ∗ϕ holds in ∗R.
A ﬁrst order statement ϕ (or ∗ϕ in ∗R) is one referring to elements (ﬁxed or variable) of R (respectively, ∗R) and to ﬁxed functions and relations on R (respectively, ∗R), that uses the usual logical connectives and (∧), or (∨), implies (→) and not (¬). Quantiﬁcation may be done over elements but not over relations or functions; that is, ∀x, ∃y are allowed, but ∀f, ∃R are not. As an example, the density of the rationals in the reals can be written as ∀x ∈ R∀y ∈ R(x < y →∃z ∈ R(z ∈ Q∧(x < z < y))), an expression meaning, “between every two reals is a rational”. From the transfer principle we can therefore immediately conclude that the statement is true in ∗R, that is, that the hyperrationals are dense in the hyperreals. The corresponding transferred statement is as follows: ∀X ∈ ∗R∀Y ∈ ∗R(X < Y →∃Z ∈ ∗R(Z ∈ ∗Q∧(X < Z < Y ))), In transferring, every set, relation and function in the original statement is replaced by its nonstandard extension, according to the ultrapower construction. We say that an element x of ∗R is ﬁnite if there is some r ∈ R such thatx < |∗r|. A simple but important theorem is the following: Theorem 2.2. If x ∈ ∗R is ﬁnite, then there is a unique r ∈ R such that x ≈ r. Any ﬁnite hyperreal is thus expressible as x = r + δ with r ∈ R and δ inﬁnitesimal. Proof. [1]Suppose x ∈ ∗R is ﬁnite. Let D1 be the set of r ∈ R such that∗ r < x and D2 the set of r0 ∈ R such that x < ∗r0. The pair (D1,D2) forms a Dedekind cut in R, hence determines a unique r0 ∈ R. A simple argument shows that |x−∗r0| is inﬁnitesimal. We call r0 in the above theorem the standard part of x and denote it as either ◦x or as st(x). Both are used, sometimes in conjunction, to improve readability. The following theorem will ﬁnd application in the next chapter. Theorem 2.3. Let (sn) be a sequence of real numbers and let l ∈ R. Then sn → l as n →∞⇐⇒ ∗sK ≈ l for all inﬁnite K ∈ ∗N. Proof. [7] Suppose that sn → l and let K ∈ ∗N be a ﬁxed inﬁnite number. We must show, for all real ε > 0, that |∗sK−l| < ε. From ordinary real analysis we know that there exists some n0 ∈ N such that ∀n ∈ N[n ≥ n0 →|sn −l| < ε]. According to the transfer principle, the following is true in ∗R: ∀N ∈ ∗N[N ≥ n0 →|∗sN −l|≤ ε].In particular, |∗sK −l| < ε as required. Conversely, suppose that ∗sK ≈ l for all inﬁnite K ∈ ∗N. For any given real ε > 0 we have ∃K ∈ ∗N∀N ∈ ∗N[N ≥ K →|∗sN −l| < ε]. By transferring this “down” to R, we get ∃k ∈ N ∀n ∈ N[n ≥ k →|sn −l| < ε]. By then taking n0 any of such extant k, we have that sn → l.

The nonstandard universe

The principles of the previous section can be used in a much broader context than just real analysis. Given any mathematical object M (whether it is a group, ring, vector space, etc.), we can construct a nonstandard version ∗M. We use a somewhat more economical construction however, by starting with a working portion of the mathematical universe V and ending up with a ∗V which will contain ∗M for every M ∈ V. This has the added advantage of preserving some of the relations between structures through the more general transfer principle. We start with the superstructure over R, denoted by V = V (R). It is deﬁned as follows:
V0(R) = R Vn+1(R) = Vn(R)∪P(Vn(R)), n ∈ N V = [ n∈N Vn(R). (P(A) denotes the power set of the set A.) If a larger (or simply diﬀerent) universe is required, start the same process with a more suitable set than R. Next one must construct a mapping ∗ : V (R) → V (∗R) associating to an M ∈ V a nonstandard extension ∗M ∈ V (∗R). The nonstandard universe can now be constructed by means of an ultrapower VN/U, and then utilising a “Mostowski collapse” [1]. This is somewhat more complicated to do than in the case of ∗R and we do not go into detail here. It is suﬃcient to consider the nonstandard universe as the set of objects ∗V = {x : x ∈∗M for some M∈ V}.
Sets in ∗V are called internal sets. It should be noted that ∗V ∈ V (∗R), but that V (∗R) contains sets that are not internal.
We now also have a transfer principle which speciﬁes which statements may be moved from one structure to the other. A bounded quantiﬁer statement is a statement which can be written so that all quantiﬁers range over a ﬁxed set. Thus, quantiﬁers like ∀x ∈ A or ∃y ∈ B are allowed, but not unbounded quantiﬁers such as∀x and∃y. Note that often boundedness is implied in the exposition and is not always speciﬁcally indicated in the statement. When given a bounded quantiﬁer statement ϕ, we obtain its nonstandard version ∗ϕ by replacing every set, function or relation in ϕ by its nonstandard counterpart. Speciﬁcally, since we can consider relations and functions as sets as well, we replace each set A by its nonstandard counterpart ∗A, whilst the logical connectives in the statement ϕ remain the same. Thus, a variable x ranging over R becomes a variable X ranging over ∗R, a function f is replaced by its extension ∗f, and a relation R is replaced by its extension ∗R.
Theorem 2.4. A bounded quantiﬁer statement ϕ holds in V if and only if ∗ϕ holds in ∗V.
The transfer principle can after some consideration be seen to apply only to internal sets. For instance, the concept of supremum implies that each bounded set will have a least upper bound. However, N seen as a member of ∗R is bounded, but has no supremum. It is therefore an external (i.e. non-internal) set. We show now that the concept of supremum transfers. The proof also provides an illustration of how to change a bounded quantiﬁer statement ϕ into ∗ϕ.
Proposition 2.5. Every nonempty internal subset of ∗R with an upper bound has a least upper bound.
Proof. The notation used in this proof refers back to our construction of the nonstandard universe. We express the fact that any nonempty subset of the standard real numbers has a least upper bound by the statement Φ(R,V2(R)) = ∀A ∈ V2(R)[A 6= ∅∧(∃x ∈ R(∀y ∈ A(y < x))) → ∃z ∈ R(∀y ∈ A(y < x)∧∀u ∈ R∀y ∈ A(y ≤ u → z ≤ u))]. Since the statement Φ = Φ(R,V2(R)) is true in V (R), the transferred ∗Φ = Φ(∗R,∗V2(R)) condition is true in V (∗R). The nonstandard version of the above statement that will hold is Φ(∗R,∗V2(R)) = ∀A ∈∗V2(R)[A 6= ∅∧(∃X ∈∗R(∀Y ∈ A(Y < X))) → ∃Z ∈∗R(∀Y ∈ A(Y < X)∧∀u ∈∗R∀Y ∈ A(Y ≤ U → Z ≤ U))]. (The capitals for the variables are not necessary and just serve to indicate that the statement is indeed nonstandard.) The transfer principle yields the following properties, which will be used later:Proposition 2.6. Let A ⊆∗R be an internal set. (i) If A contains arbitrarily large ﬁnite numbers, then it also contains an inﬁnite number.
(ii) If A contains arbitrarily small positive inﬁnite numbers, then it contains a positive ﬁnite number.
These two are known as the overﬂow and underﬂow properties, respectively. We give the proof as another illustration of the use of the Transfer Principle. Proof.
(i) Since A is an internal set, if it has an upper bound, it must have a least upper bound. However, if it did not contain an inﬁnite number, it would be bounded by any inﬁnite number. Such a number would necessarily be inﬁnite, leading to a contradiction.
(ii) The same type of proof as in (i) holds here, once it is recognised that the transfer principle guarantees that an internal set bounded from below has an inﬁmum.
Note that these properties are also easily obtained from the ultraﬁlter construction. By taking reciprocals, similar properties can be seen to hold for inﬁnitesimals. An important property of any nonstandard universe constructed as an ultrapower is that of ℵ1-saturation: Proposition 2.7. If (Am)m∈N is a countable decreasing sequence of nonempty internal sets, then ∩m∈NAm 6= ∅. A useful reformulation of this is known as countable comprehension: Given any sequence (An)n∈N of internal subsets of an internal set A, there is an internal sequence (An)n∈∗N of subsets of A that extends the original sequence. This property will be used in the construction of Loeb measure.

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Nonstandard topology

Before doing analysis in a nonstandard universe, we must clearly understand the topology. Firstly, we see that the concept of being inﬁnitely close, and therefore the idea of a monad, can be extended:
Deﬁnition 2.2. Let (X,τ) be a topological space. (i) For a ∈ X the monad of a is monad(a) = \ a∈U∈τ∗U.
(ii) For x ∈ ∗X, we write x ≈ a if x ∈ monad(a). (iii) x ∈∗X is said to be nearstandard if x ≈ a for some a ∈ X. (iv) For any Y ⊆ ∗X, we denote the nearstandard points in Y by ns(Y ). (v) st(Y ) = {a ∈ X : x ≈ a for some x ∈ Y} is called the standard part of Y . The following result allows us to generalise the pointwise standard part mapping:
Proposition 2.8. A topological space X is Hausdorﬀ if and only if monad(a)∩monad(b) = ∅ for a 6= b, a,b ∈ X. This means we can deﬁne the function st : ns(∗X) → X as st(x) = the unique a ∈ X with a ≈ x. Again, we use the notation ◦x = st(x) interchangeably. We mention some general topological results. Proposition 2.9. Let (X,τ) be separable and Hausdorﬀ. Suppose Y ⊆ ∗X is internal and A ⊆ X. Then (i) st(Y ) is closed, (ii) if X is regular and Y ⊆ ns(∗X), then st(Y ) is compact, (iii) st(∗A) = A (closure of A), (iv) if X is regular, then A is relatively compact iﬀ ∗A ⊆ ns(∗X). Since we will be dealing almost exclusively with continuous functions, we should introduce corresponding notions in the nonstandard universe. Deﬁnition 2.3. Let Y be a subset of ∗X for some topological space X and let F : ∗X → ∗R be an internal function. Then F is said to be S-continuous on Y if for all x,y ∈ Y we have x ≈ y ⇒ F(x) ≈ F(y). The following result allows us to switch from the one notion of continuity to another. Theorem 2.10. If F : ∗X → ∗X is S-continuous on an interval ∗[a,b] for real a,b and F(x) is ﬁnite for some x ∈ ∗[a,b], then the standard function deﬁned in [a,b] by f(t) =◦F(t) is continuous and ∗f(τ) ≈ F(τ) for all τ ∈ ∗[a,b]. Given a real function f deﬁned on an interval [a,b], we shall call any function F on ∗[a,b] such that f(t) =◦F(t), a lifting of f.

Loeb measure

A Loeb measure is a standard measure, but constructed from a nonstandard one. That is, the Loeb measure exists on a σ-algebra and obeys all the usual rules for a measure, for example countable additivity. We start with a given internal set Ω, an algebra A of internal subsets of Ω and a ﬁnite internal ﬁnitely additive measure µ on A. Thus µ is a function from A to ∗[0,∞) such that µ(Ω) < ∞ and µ(A ∪ B) = µ(A) + µ(B) for disjoint A,B ∈ A. (We focus only on bounded Loeb measures; inﬁnite ones shall not concern us in the sequel.) We can then deﬁne the mapping ◦µ : A→ [0,∞) by ◦µ(A) =◦(µ(A)). This is ﬁnitely additive and therefore (Ω,A,µ) is a standard ﬁnitely additive measure space. This is not usually a measure, since ◦µ is not always σ-additive. We shall see shortly, however, that it is almost a measure. The following crucial theorem was proved by Loeb [27]. It is possible to give a quick proof using Caratheodory’s extension theorem, but we shall follow Cutland [7] and give a more straightforward approach. Theorem 2.11. There is a unique σ-additive extension of ◦µ to the σ-algebra σ(A) generated by A. The measure theoretic completion of this measure is the Loeb measure associated with µ, denoted by µL. The completion of σ(A) is the Loeb σ-algebra, denoted by L(A). The more straightforward proof depends on the notion of a Loeb null set: Deﬁnition 2.4. Let B ⊆ Ω, where B is not necessarily internal. We call B a Loeb null set if for each standard real ε > 0 there is a set A ∈A with B ⊆ A and µ(A) < ε. This allows us to make precise the notion that A is almost a σ-algebra: Lemma 2.12. Let (An)n∈N be an increasing family of sets, with each An in A and let B =Sn∈N An. Then there is a set A ∈A such that (i) B ⊆ A (ii) ◦µ(A) = limn→∞ ◦µ(An) and (iii) A\B is null Proof. Let α = limn→∞ ◦µ(An). For any ﬁnite n, µ(An) ≤ ◦µ(An) + 1 n ≤ α + 1 n . Let (AN)N∈∗N be a sequence of sets in Aextending the sequence (An)n∈N, made possible by ℵ1 saturation (see 2.2). The overﬂow principle then guarantees an inﬁnite N such that
µ(AN) ≤ α 1 N
If we now let A = AN, (i) will hold because An ⊆ A for each n. Also, µ(An) ≤ µ(A) for ﬁnite n, so ◦µ(An) ≤◦ µ(A) ≤ α and therefore ◦µ(A) = α. This gives (ii). For (iii), note that A\B ⊆ A\An and ◦µ(A\An) = ◦µ(A)−◦µ(An) → 0 as n →∞. Thus A is a σ-algebra modulo null sets. We can now deﬁne the concepts Loeb measurable and Loeb measure exactly: Deﬁnition 2.5. (i) Let B ⊆ Ω. We say that B is Loeb measurable if there is a set A ∈A such that A4B (the symmetric diﬀerence of A and B) is Loeb null. The collection of all the Loeb measurable sets is denoted by L(A). The algebra L(A) is known as the Loeb algebra. (ii) For B ∈ L(A) deﬁne µL(B) = ◦µ(A) for any A ∈A with A4B null. We call µL(B) the Loeb measure of B. This brings us to the central theorem of Loeb measure theory. Theorem 2.13. L(A) is a σ-algebra and µL is a complete σ-additive measure on L(A). The measure space Ω = (Ω,L(A),µL) is called the Loeb space given by(Ω ,A,µ). If µ(Ω) = 1, we refer to Ω as a Loeb probability space.

Loeb counting measure

We devote a short but separate section to the idea of counting measures because the idea is prominent throughout the sequel. Let Ω = {1,2,…,N}, where N ∈ ∗N\N. The set Ω is internal. Deﬁne the counting probability ν on Ω by ν(A) = |A| N , for A ∈ ∗P(Ω) = A. The cardinality function |·| transfers, so |A| can be interpreted as an extension of ﬁnite, standard cardinality. The Loeb counting measure νL is the completion of the extension to σ(A) of the ﬁnitely additive measure ◦ν. An easy but important example of the use of such a counting measure is in the construction of Lebesgue measure. Deﬁnition 2.6. Fix N ∈ ∗N\N and let Mt = N−1. The hyperﬁnite time line for the interval [0,1] based on the inﬁnitesimal Mt is the set T = {0,Mt,2Mt,3Mt,…,1−Mt}. The following theorem provides an intuitive construction of Lebesgue measure.

Contents
1 Introduction to Brownian motion and Hausdorﬀ dimension
1.1 Properties of Brownian motion
1.2 Hausdorﬀ dimension
1.3 Some fractal properties of Brownian paths
2 Introduction to nonstandard analysis and Loeb measure theory
2.1 The hyperreals
2.2 The nonstandard universe
2.3 Nonstandard topology
2.4 Loeb measure
2.5 Loeb counting measure
3 A nonstandard version of Hausdorﬀ dimension
4 Some applications to the fractal geometry of Brownian motion
4.1 Anderson’s construction of Brownian motion
4.2 Brownian local time
4.3 The image of a set under Brownian motion
5 Nonstandard analysis of rapid points of functions
6 The Fourier dimension of rapid points
6.1 Introduction
6.2 Large increments of Brownian motion
7 Appendix

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The fractal geometry of Brownian motion