Limiting Probability Measures

The coordinates along any fixed direction(s), of points on the sphere $S^{n-1}(\sqrt{n})$, roughly follow a standard Gaussian distribution as $n$ approaches infinity. We will revisit this classical result from a nonstandard perspective. We first develop a general theory for the asymptotic behavior of integrals over varying domains that are increasing in dimension. We then define an appropriate surface area measure on the hyperfinite-dimensional sphere $S^{N-1}(\sqrt{N})$ and show that for any Gaussian-integrable function $f$, the corresponding Loeb integral of $\bf{st}({^*}f)$ is equal to the expectation of $f$. We also show that this common value is a limit of spherical integrals of $f$ over $(S^{n-1}(\sqrt{n}))_{n \in \mathbb{N}}$. A review of the requisite nonstandard analysis is provided.


Overview
The first section reviews basic nonstandard analysis. The aim is to make the rest of the paper accessible to people familiar with probability theory (at the level of a graduate book such as [3]). The interested reader is directed toward the numerous good books on nonstandard analysis for more background (see [1], [6], and [15]).
The second section studies when a Loeb measure on a higher dimensional nonstandard space contains information about all events on a given standard probability space. Section 3 contains results concerning when a limit of integrals on varying measure spaces is given by a Loeb integral on a single limiting measure space.
In Section 4, we prove a general formulation of a classical result (going back to Boltzmann [4], Maxwell [11] and Poincaré [14]) on high-dimensional spheres. Classically (see Proposition 4.1), the proportion of points on S n−1 ( √ n) whose first k coordinates lie in a given Borel subset B of R k converges to µ(B), where µ is the standard Gaussian measure on R k . We generalize this to a result (see Theorem 4.5) showing that the limit of certain spherical integrals of a function is the same as its integral with respect to µ. Toward that end, we also obtain some necessary and sufficient conditions for a function to be Gaussian integrable (see Theorem 4.5).
The connection between surface area on high-dimensional spheres and the Gaussian measure was used by Lévy [10] to do infinite dimensional analysis, and later by Wiener [19] to construct a Brownian motion. McKean [12] surveyed most of the relevant work from that period. With access to spheres in hyperfinite dimensions, nonstandard analysis allows a natural framework within which to study these topics. This was exploited by Cutland and Ng in [7] to give a new construction of the Wiener measure.
Our machinery is naturally applicable to slices of high-dimensional spheres and the Gaussian Radon transform, as studied by Sengupta [18]. These ideas will be used in an upcoming manuscript to generalize Sengupta's result [18,Theorem 2.1] in the spirit of the recent work of Peterson and Sengupta [13]. 1. A non-logical crash course in nonstandard analysis 1.1. Basic nonstandard analysis. There are many approaches to nonstandard analysis, eight of which were described in [2]. We follow the superstructure approach, as done in [1]. Roughly, a nonstandard extension of a set X (consisting of atoms or urelements i.e., we view each element of X as an "individual" without any structure, set-theoretic or otherwise) is a superset * X that preserves the "firstorder" properties of X. That is, a property which is expressible using finitely many symbols without quantifying over any collections of subsets of X is true if and only if the same property is true of * X. This is called the transfer principle (or just transfer for brevity). The set * X should contain, as a subset, * Y for each Y ⊆ X.
Like subsets, other mathematical objects defined on X also have extensions. So, a function f : X → Y extends to a map * f : * X → * Y , and relations on X extend to relations on * X. Hence there is a binary relation * < on * R, which we still denote by < (an abuse of notation that we will frequently make), and which is the same as the usual order when restricted to R. Thus, * R is an ordered field of characteristic 0. Indeed all (the infinitely many) axioms for ordered fields of characteristic 0 hold for it by transfer. The symbols in a sentence such as "∀x > 0 ∃y(x = y 2 )" (which is expressing the proposition that each positive number has a square root) have new meanings in the nonstandard universe: by " < ", we are now interpreting the extension of the order on R. Yet the sentence is true in * R by transfer! A non-first-order property of X may not transfer to * X. For instance, we shall see (cf. Proposition 1.2) that any "non-trivial" extension of R contains infinite elements (i.e., those that are larger than all real numbers in absolute value), as well as infinitesimal elements (i.e., those that are smaller than all positive real numbers in absolute value). Thus, the Archimedean property of R does not transfer. The set of finite nonstandard real numbers, denoted by * R fin , is a subring of the non-Archimedean field * R. To see what went wrong, note that the following sentences formally express the Archimedean property for R and its transfer, respectively: The transferred sentence (1.2) no longer expresses the Archimedean property (though it still expresses an interesting fact about * R). The issue is that we are only able to quantify over * N (and not on N) after transfer. To keep quantifying over N, we would have to transfer an "infinite statement" (saying that for every x, either 1 > x, or 2 > x, or 3 > x, or . . .), which is not a valid first-order sentence.
Another non-example is the least upper bound principle: the set N, viewed as a subset of * R, is bounded (by any positive infinite element), yet has no least upper bound (as any upper bound minus one is also upper bound). The issue here is that the least upper bound property for R is expressed via the second-order statement: One way to express this as a first-order statement is to quantify over the powerset, P(R), of R. If our nonstandard map * was able to extend sets of subsets of X as well, then the above would transfer to the following *-least upper bound property: This would imply that N would not be an element of * P(R), whatever the latter object is (the object * P(R) would in fact be a subset of P( * R) due to the transfer of the sentence "∀A ∈ P(R) ∀x ∈ A (x ∈ R)"). As we shall see, we do indeed extend P(X). An element of * P(X) is called an internal subset of * X. The previous example leads to the observation that * P(R) is not a superset of P(R) in the literal sense. It does, however, contain as an element the extension * A for any A ∈ P(R).
In general, we fix a set X consisting of atoms, and extend what is called the superstructure V (X) of X, which is defined inductively as follows: By choosing X suitably, the superstructure V (X) can be made to contain all mathematical objects relevant for a given theory. For example, if R ⊆ X, then all collections of subsets of R, including all topologies on R, all sigma-algebras on R, etc., live as objects in V 2 (X) ⊆ V (X). For a finite subset consisting of k objects from V m (X), the ordered k-tuple of those objects is an element of V n (X) for some larger n (and hence the set of all k-tuples of objects in V m (X) lies as an object in V n+1 (X)). For example, if x, y ∈ V m (X), then the ordered pair (x, y) is just the set {{x}, {x, y}} ∈ V m+2 (X). Identifying functions and relations with their graphs, V (X) also contains, if R ⊆ X, all functions from R n to R, all relations on R n , etc., for all n ∈ N.
We extend the superstructure V (X) via a nonstandard map, which, by definition, is any map satisfying the following axioms: (NS1) The transfer principle holds. (NS2) * α = α for all α ∈ X.
(NS3) { * a : a ∈ A} * A for any infinite set A ∈ V (X). A nonstandard map may not be unique. In practice, however, we fix a standard universe V (X) and a nonstandard map * . The reader is referred to [5,Theorem 4.4.5,p. 268] or [1, Chapter 1] for a proof of existence of a nonstandard map.
An object that belongs to * A for some A ∈ V (X) is called internal. We have already seen several examples of internal sets and functions: * N, * R, * f (for any standard function f ), etc. Unlike these examples, (NS3) guarantees the existence of internal objects that are not * α for any α ∈ V (X). Internal objects are those that inherit properties from their standard counterparts by transfer. Thus, for example, the transfer of Archimedean property (see (1.2)) says that * N does not have an upper bound. Note that, by transfer, the class of internal sets is closed under Boolean operations such as finite unions, finite intersections, etc. Definition 1.1. For a cardinal number κ, a nonstandard extension is called κsaturated if any collection of internal sets that has cardinality less than κ and that has the finite intersection property has a non-empty intersection.
We will henceforth assume that the nonstandard extension we work with is sufficiently saturated (cf. [ Proof. Any element in the non-empty intersection ∩ n∈N {x ∈ * R : x > n} must be infinite. The multiplicative inverse of any infinite element is infinitesimal. The next result holds since elements of * N are at least one unit apart (due to transfer) and any finite element of * N has a least natural number larger than it. Proposition 1.3. Any N ∈ * N\N is infinite. We express this by writing N > N.
Any N > N is said to be hyperfinite. The following result is a consequence of the fact that N is not internal. See also [1, Proposition 1.2.7, p.21].
The next result says that one can think of a finite nonstandard real number z as having a real part, and an infinitesimal part (in fact, this real part is just sup{y ∈ R : y ≤ z}). See [8, Theorem 2.10, p. 55] for a proof.
The next result gives a nice characterization of limit points of sequences (see [ The following consequence of saturation (see [1, Lemma 3.1.1, p. 64] for a proof) will be useful in the sequel: Proposition 1.7. A countable union of disjoint internal sets is internal if and only if all but finitely many of them are empty.

Loeb Measures.
Let Ω be an internal set in a nonstandard universe * V (X). Let F be an internal algebra on Ω, i.e., an internal set consisting of subsets of Ω that is closed under complements and finite unions. Given a finite, finitely additive internal measure P (i.e., P : F → * R ≥0 satisfies P(∅) = 0, P(Ω) < ∞, and P(A ∪ B) = P(A) + P(B) whenever A ∩ B = ∅), the map st(P) : F → R ≥0 is an ordinary finite, finitely additive measure. By Proposition 1.7, it follows that st(P) satisfies the premises of Carathéodory Extension Theorem. By that theorem, it extends to a unique measure on σ(F ) (the smallest sigma algebra containing F ), whose completion is called the Loeb measure of P. The corresponding complete measure space (Ω, L(F ), LP) is called the Loeb space of (Ω, F , P).
We will use the following simplification of [16, Theorem 5.1, p. 105] extensively: Proposition 1.8. Let (Ω, L(F ), LP) be the Loeb probability space of (Ω, F , P). Suppose F : Ω → * R is an internal function that is measurable in the sense that For a standard measure space (Ω, F ), let Prob(Ω, F ) be the set of probability measures on (Ω, F ). If C ∈ V (X) is a collection of measure spaces, then Prob(C) denotes the set of all probability measures on elements in C. Any element in * Prob(C) is a finitely additive internal probability on an internal measure space. For any P ∈ Prob(C), there is an integral operator that takes certain functions (those in the space L 1 (P) of integrable real-valued functions on the underlying sample space of P) to their integrals with respect to P. Thus if (Ω, F , P) ∈ * C is an internal probability space, we also have the associated space * L 1 (Ω, P) of * -integrable functions.
For any * -integrable F : Ω → * R, one then has * ˆΩ F dP ∈ * R, which we call the * -integral of F over (Ω, P). This * -integral on * L 1 (Ω) inherits many properties (an important one being linearity) from the ordinary integral by transfer. If F is finite almost surely with respect to the corresponding Loeb measure, then st(F ) is Loeb measurable by Proposition 1.8. In that case, it is interesting to study the relation between the * -integral of F and the Loeb integral of st( * F ). The following result covers this for a useful class of functions (see [16, Theorem 6.2, p.110] for a proof): Then the following are equivalent: (1) * ˆΩ |F | dP ∈ * R fin ; and st * ˆΩ (3) * ˆΩ |F | dP ∈ * R fin ; and for any B ∈ F we have: A function satisfying the conditions in Theorem 1.9 is called S -integrable on (Ω, F , P). Given a Loeb measurable f : Ω → R, a natural question is when does it occur as the standard part of an internal function. An internal measurable function The following theorem shows that * -integrable functions can be characterized as those possessing S-integrable liftings (see [16,Theorem 6.4,p.111] for a proof).

On sections of some Loeb measures
Let X be a set of urelements, and its superstructure V (X) be defined inductively as in (1.3). In this paper, we work with measures defined on a sequence of measure spaces, and want to construct a natural Loeb measure on any element in the nonstandard extension of such a sequence. One issue in doing so could be that the measure spaces might not all lie in V n (X) for a single n ∈ N (in which case, we cannot think of the sequence of measure spaces as an element of V (X)).
In particular, this would be an issue if our measure spaces were the Borel spaces (R n , B(R n )) and X was the set of real numbers. To get around this difficulty, one could take a set X that contains (copies of) R n for each n ∈ N. In what follows, there will be a set E such that we assume X to contain copies of E n for all n ∈ N.
If ν is a measure on a subset Ω of E n for n ∈ N ≥k and B is a measurable subset of E k , then we denote ν(Ω ∩ (B × E n−k )) by ν(B). Similarly, we can talk about integrating a function f : E k → R over Ω by extending f canonically to E n . Proposition 2.1. Let Ω ∈ * V (X) be such that Ω ⊆ * E N for some N ∈ * N. Let E be a sigma-algebra on E, and let E k denote the corresponding product sigma algebra on E k for each k ∈ N. Let * E N denote the corresponding internal algebra on * E N (defined by extension of the sequence {E k } k∈N , which is an element of V (X) when viewed as a function on N). Let F be the restriction of * E N to Ω.
Fix k ∈ N and suppose P ∈ Prob(E k , E k ). Let ν ∈ * Prob(Ω, F ). If Lν is the corresponding Loeb measure, and if N ≥ k, then: Proof. For ǫ ∈ R >0 , find n ǫ ∈ N such that for any n ∈ N ≥nǫ , we have |f n (x) − f (x)| < ǫ on E k . By transfer, for all n ∈ N ≥nǫ , we get is arbitrary, this impliesˆΩ st( * f )dLν =ˆE k f dP, as desired.
Corollary 2.2. In the above setting, if P is a Radon probability measure on E k , then the following are equivalent: Proof. (1) ⇔ (2) follows from Proposition 2.1. Also, (2) ⇒ (3) is immediate. Further, (3) ⇔ (4) holds by the following argument for a closed set C := E k \O: Thus it suffices to prove that (3) and (4) together imply (2). To that end, assume (3) and (4) Proof. If B n := {x ∈ E k : |f (x)| < n} for n ∈ N, then the required probability is completing the proof.

On the long-term behavior of a sequence of measure spaces
Let {(Ω n , F n , ν n )} n∈N be a sequence of probability spaces. Viewing the sequence as a function on N, we get an internal probability space (Ω N , F N , ν N ) for each N > N. Note that we have been dropping the * when it is clear from context that the index N is hyperfinite. Philosophically, the Loeb space (Ω N , L(F N ), Lν N ) for N > N should capture the long-term behavior of the sequence {(Ω n , F n , ν n )} n∈N of probability spaces. The following theorem makes this more precise in some cases.
Theorem 3.1. Suppose that E ∈ V (X) and k ∈ N are such that for all large n, we have Ω n ⊆ E n ′ for some n ′ ≥ k, where n ′ depends on n. Suppose E is a sigma algebra on E, and that F n , the given sigma-algebra on Ω n , is induced by the product Then, f is integrable over (Ω n , ν n ) for large n, so that the sequence α f,n :=ˆΩ Proof. For a fixed ǫ ∈ R >0 , there exists ℓ ǫ ∈ N such that the following holds: for any m ≥ ℓ ǫ , there is an n ǫ,m ∈ N such that for all n ≥ n ǫ,m , we havê Ωn∩{|f |≥m} |f | dν n < ǫ.
In particular, f is integrable on Ω n for all n > n ǫ,ℓǫ , with the integral of the absolute value being at most (ℓ ǫ + ǫ). Further, for any M, N > N, transfer yields * ˆΩ Given N > N, * f is S-integrable on Ω N by Theorem 1.9 (2), while α f,N is the *integral of * f over (Ω N , ν N ) by transfer. Theorem 1.9(4) now finishes the proof.
Let Ω n , E, and E be as in Theorem 3.1. Let P be a Radon probability measure on E k such that Lν N ( * B) = P(B) for any B ∈ E k and N > N.
(ii) If f : E k → R is bounded and measurable, then We are interested in finding conditions that make the result in (ii) true for all µ-integrable functions. The following result provides some answers.  Since * f is S-integrable on Ω N0 , it follows that st( * f ) is Loeb integrable on Ω N0 . Hence the limit on the right side of (3.2) is zero, as desired.
We now assume that f : R k → R is such that the double-limit in (3) is zero. Fix as m → ∞, Lν N -almost surely on Ω N . By monotone convergence theorem, In general, for a function f : E k → R (not necessarily satisfying the conditions in Theorem 3.4), the following lemma allows us to still approximate its integral by a suitably modified sequence of integrals over (Ω n , ν n ).
Lemma 3.5. In the setting of Corollary 3.3, let f : E k → R be P-integrable. Given any ǫ, δ, θ ∈ R >0 there exist an n 0 ∈ N and functions g n : Ω n → R for all n ∈ N ≥n0 such that the following hold:

Proof. By Corollary 3.3[(iii)], we know that
Thus, for any N > N, the map st( * f ) is Loeb integrable on Ω N , and hence has an S-integrable lifting G N : Ω N → * R by Theorem 1.10. In particular, Without loss of generality, we can assume that |G N | ≤ N for all N > N (as we may replace G N by the function min{G N , N } which still satisfies (3.3) and (3.4)). Thus, for the given ǫ, δ, θ ∈ R >0 , the following internal set contains * N\N.
We can strengthen Lemma 3.5 by requiring the functions to be defined on E k : Theorem 3.6. In the setting of Proposition 3.3, let f : E k → R be P-integrable. Given any ǫ, δ, θ ∈ R >0 there exist an n 0 ∈ N and functions g n : E k → R for all n ∈ N ≥n0 such that the following hold: For any bounded measurable g : E k → R, expressing g as a uniform limit of simple functions yieldsˆΩ n gdν n =ˆE k gdν ′ n . Let S 0 be the set of all spheres centered at origin (in any dimension n ∈ N snd of any radius r ∈ R >0 ). More formally, we have: Since X contains copies of all Euclidean spaces, S 0 is an element of V (X). It is known that there is a unique rotation-preserving probability measure on any sphere centered at origin equipped with its Borel sigma-algebra. More formally: ∀S ∈ S 0 ∃!σ ∈ Prob((S, B(S)) (∀R ∈ Rot(S) ∀A ∈ B(S) [σ(R(A)) =σ(A)]).
For any S ∈ * S 0 in the nonstandard universe, the transfer principle implies that the set * Prob(S, * B(S)) consists of a unique finitely additive internal function, saȳ σ S : * B(S) → * [0, 1], that is * -rotation preserving andσ S (S) = 1. By the usual Loeb measure construction, we get L(σ S ) on L( * B(S)) (a sigma algebra containing σ( * B(S)), which we call the uniform Loeb surface measure on S.
In finite dimensions, we also have the notion of surface area. For each S ∈ S 0 , one can consider the surface area map σ S : B(S) → R, which satisfies the following: • For any d ∈ N and any a ∈ R >0 , we have σ S d (a) (S d (a)) = c d · a d , where • For any S ∈ S and any A ∈ B(S), we haveσ S (A) = σ S (A) σ S (S) .
By transfer, we have the notion of * -surface area in the nonstandard universe. This could be used as an alternative way to define the uniform Loeb surface measure.
For n ∈ N, let S n be the sphere S n−1 ( √ n). We useσ n to denoteσ Sn . Fix k ∈ N and let µ denote the standard k-dimensional Gaussian measure. Let B k (a) denote the open ball of radius a in R k . For a set B ∈ B(R k ) and any n ∈ N ≥k , we defineσ n (B) to be the value ofσ n ({x ∈ S n : π k (x) ∈ B}) =σ n (B × R n−k ) ∩ S n , where π k is the projection onto R k . Similarly, a function f : R k → R is canonically extended to R n by using "f (x, y)" to denote f (x) for all x ∈ R k and y ∈ R n−k . In this setting, we first reprove the classical result of Maxwell, Boltzmann, and Poincaré (see [11], [4], and [14] respectively): Proof. By Proposition 1.6, it suffices to prove the first assertion. Let λ denote the Lebesgue measure on R k . By "Sengupta's disintegration formula" (see [18,Proposition 4.1]), we have, for any f : R k → R, = a n,k b n,kˆR where a n,k = Γ n 2 Γ n−k 2 · n−k Note that lim n→∞ a n,k = lim n→∞ b n,k = 1 for all k ∈ N (the first limit following from Stirling's formula, see [17, equation 103, p. 194]). If f (x) = ½ B for some B ∈ B(R k ), then for large values of n, the integrand is bounded by which is integrable on R k . Thus, in that case, the integral in (4.1) converges to µ(B) as n → ∞ by the dominated convergence theorem.
Corollary 4.2. For k ∈ N and N > N, almost all points on S N have finite first k coordinates. That is,  dλ(x) = 0.
Since ǫ ∈ R >0 is arbitrary, equation (4.1) yields For all n ∈ N ≥2(k+2) , the sequence in the statement of the Corollary is bounded above by (a constant times) the sequence in (4.2), thus completing the proof.
Lemma 4.4. There is an n ′ ∈ N and a C ∈ R >0 such that for all g ∈ L 2 (R k , µ): Sn |g| dσ n ≤ C E µ (|g|) + E µ (g 2 ) for all n ∈ N ≥n ′ . Proof. Fix g ∈ L 2 (R k , µ). Using (4.1), we see thatˆS n |g| dσ n is at most equal tô where C ′ is an absolute constant bounding |a n,k b n,k |. .
Theorem 4.5. The following are equivalent: (1) A function f : R k → R is integrable with respect to µ.
(2) The nonstandard extension * f is S-integrable on S N for all N > N.
In that case, we have lim n→∞ˆS n f dσ n =ˆR k f dµ.

Furthermore, this limit is equal toˆS
N st( * f )dLσ N for any hyperfinite N .
Let n ′ be as in Lemma 4.4. For any m, m ′ ∈ N with m ′ > m, and n > n ′ , we have: .