Non-Standard Probability

Non-standard analysis promises, at several points, that hyperfinite methods give "slick constructions of Lebesgue measure, Brownian motion, and Itô integration" — applications.md and nonstandard-integration.md. This page redeems those promises and explains what the approach does and does not buy.

The central construction is Loeb measure: an internal, finitely additive hyperreal-valued measure is converted into a genuine, countably additive standard measure by taking standard parts. The result is that a hyperfinite probability space — a space with an infinite-but-hyperfinite number of equally likely points — carries, after Loeb's construction, every standard probability space of interest.

References: Loeb (1975); Albeverio–Fenstad–Høegh-Krohn–Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics; Nelson, Radically Elementary Probability Theory.

1. Hyperfinite Probability Spaces

Fix an infinite hypernatural $N\in {}^{\ast }\mathbb{N}\setminus \mathbb{N}$ (hyperreals.md) and let

$$\Omega =\{1,2,\dots ,N\}\subseteq {}^{\ast }\mathbb{N},$$

a hyperfinite set: infinite, but internally finite, so every theorem about finite sets transfers to it (transfer-principle.md). Put $\mathcal{A}$ = the internal subsets of $\Omega$ and

$$P_0(A)=\frac{|A|}{N}\in {}^{\ast }[0,1],$$

where $|A|$ is the internal cardinality. By transfer, $P_0$ is an internally countably additive probability — the uniform distribution on a finite set, seen through the transfer principle.

This is already the object that finitely additive probability was reaching for (finite-additivity.md §2): a genuinely uniform distribution on an infinite set, with each point receiving the infinitesimal mass $1/N$. No standard countably additive measure can do this, and here it comes for free.

The catch is that $P_0$ takes values in ${}^{\ast }[0,1]$, and is only internally countably additive: a genuinely countable union of internal sets is usually external and outside $\mathcal{A}$. Loeb's theorem repairs exactly this.

2. The Loeb Measure

Theorem (Loeb, 1975). Let $(\Omega ,\mathcal{A},P_0)$ be an internal finitely additive probability space. Then $\mathrm{st}\circ P_0:\mathcal{A}\to [0,1]$ extends uniquely to a countably additive probability measure $P_L$ on $\sigma (\mathcal{A})$, and its completion is the Loeb space $(\Omega ,{\mathcal{A}}_L,P_L)$.

Proof idea. $\mathrm{st}\circ P_0$ is finitely additive on the algebra $\mathcal{A}$. To apply Carathéodory (measure-theory.md §5) one needs continuity at $\varnothing$, and this comes free from saturation: a decreasing sequence of internal sets with empty intersection must have an empty member at some finite stage, since an internal sequence with the finite intersection property has nonempty intersection. $\square$

The one-line summary. ${\aleph }_1$-saturation converts finite additivity into countable additivity. Everything that finitely additive probability could not do (finite-additivity.md §3) becomes available, because the countable additivity is recovered after the standard-part map — the infinitesimal leakage that breaks continuity in the standard finitely additive theory is annihilated by $\mathrm{st}$.

3. What It Constructs

Lebesgue measure. Take $\Omega =\{k/N:k=1,\dots ,N\}$, the hyperfinite grid in ${}^{\ast }[0,1]$, with the uniform internal measure. The standard-part map $\mathrm{st}:\Omega \to [0,1]$ is Loeb-measurable, and the pushforward of $P_L$ under $\mathrm{st}$ is Lebesgue measure on $[0,1]$. So ${\lambda }^1$ is literally the counting measure on an infinitesimally fine grid, viewed from the standard world. This is the sense in which the physicist's "sum over infinitesimal cells" picture is correct.

Brownian motion (Anderson, 1976). Let $\Delta t=1/N$ and take the hyperfinite random walk

$$X_k=\sum _{j=1}^k{\epsilon }_j\sqrt{\Delta t},{\epsilon }_j=\pm 1\text{ internally i.i.d.},$$

on the hyperfinite coin-flip space $\{-1,+1{\}}^N$. Then $B_t=\mathrm{st}(X_{\lfloor Nt\rfloor })$ is, $P_L$-a.s., a standard Brownian motion. All of brownian-motion.md §2 — Kolmogorov extension, continuity criteria, tightness and Prokhorov — is replaced by a random walk with infinitesimal steps, which is exactly the informal picture that motivates the object.

The Itô integral becomes a hyperfinite sum ${\sum }_k{H}_{t_k}(X_{k+1}-X_k)$, internally an ordinary finite sum. The left-endpoint evaluation that defines Itô (as opposed to Stratonovich) is visible as a choice about which grid point to sample, and $(dB{)}^2=dt$ becomes the internal identity ${\sum }_k(\Delta X_k{)}^2=N\cdot \Delta t=1$.

Other constructions. Haar measure on compact groups, Feynman-style path integrals with hyperfinite time slicing, and existence results for stochastic differential equations all admit hyperfinite treatments in which the object is constructed by transfer from the finite case.

4. Infinitesimal Probabilities and Regularity

A separate motivation, independent of the constructions above, is the demand that possible events not receive probability zero. In a standard model, picking a uniform point of $[0,1]$ gives every individual outcome probability exactly $0$, even though each is possible; in a hyperfinite model each grid point has probability $1/N>0$, an infinitesimal.

Regularity. A probability function is regular if $P(A)=0$ only for $A=\varnothing$. Standard real-valued probability cannot be regular on an infinite space (uncountably many disjoint positive values would sum past 1); hyperreal-valued probability can.

What this does not deliver:

• The comparisons are model-dependent. The value $1/N$ depends on the choice of $N$ and of the ultrafilter used in the ultrapower construction, and different choices give incomparable answers to questions like "is the probability of this event twice that one?" for infinitesimal events. The construction is non-constructive at exactly the point where non-measurable sets are.
• Conditioning on infinitesimals is still ambiguous. $P(A\mid B)$ with infinitesimal $P(B)$ is now well defined as a hyperreal ratio, but its standard part depends on how $B$ is represented internally — the Borel–Kolmogorov §6 ambiguity is relocated, not removed.
• Nothing standard is proved that could not be proved standardly. By the transfer principle and conservativity of the non-standard framework over the standard one, every standard theorem obtained by these methods has a standard proof. The gain is in construction and intuition, not in strength.

5. Assessment

Non-standard probability is best understood as a construction technique and an alternative foundation for the same theory, not a rival to the Kolmogorov axioms: the output of Loeb's theorem is an ordinary countably additive probability space, and the standard theorems then apply verbatim. Its distinctive contributions are:

1. Every space is hyperfinite. By Anderson's and Loeb's results, any standard (Radon) probability space is the image of a hyperfinite one. Combinatorial arguments valid for finite spaces transfer, which is why existence proofs shorten so dramatically.
2. Infinite objects get finite proofs. Brownian motion is a random walk; Lebesgue measure is counting; the Itô integral is a sum. The idealizations of §3 are the informal derivations of physics, made rigorous rather than replaced.
3. Regularity is available, which is what finite-additivity.md wanted without giving up $\sigma$-additivity — at the price of a non-constructive choice of model.

6. Where This Is Used

• Non-standard analysis, especially applications.md and nonstandard-integration.md — the forward promises this page redeems.
• Countable vs. finite additivity §5 — the alternative route to uniform-on-infinite-set distributions.
• Brownian motion — the standard construction, for comparison.
• Measure theory §5 — Carathéodory, used inside Loeb's proof.