Borel, Wiener, Poisson, Kuratowski, etc.
These notes separate three layers:
- measurable-space structure (Borel sigma algebras, standard Borel spaces),
- probability measure on that measurable space (Wiener measure, Poisson law),
- filtration/time structure (adaptedness, stopping times, predictability, inaccessible times, martingale continuity).
1 Definition of Polish space
A Polish space is a topological space \(X\) such that:
- \(X\) is separable (it has a countable dense subset), and
- \(X\) is completely metrizable (there exists a metric \(d\) inducing the topology for which \((X,d)\) is complete).
Examples:
- \(\mathbb{R}^n\) with the usual topology,
- any separable Banach space (e.g. \(C([0,1])\) with the sup norm),
- the separable Hilbert space \(\ell^2\),
- Cantor space \(2^{\mathbb{N}}\) with the product topology,
- Baire space \(\mathbb{N}^{\mathbb{N}}\) with the product topology,
- \(C([0,\infty),\mathbb{R})\) with the topology of uniform convergence on compact sets,
- \(D([0,\infty),\mathbb{R})\) with the Skorokhod \(J1\) topology.
Remarks:
- Polish is a topological notion.
- A given set can carry many different Polish topologies.
- Different Polish topologies can induce the same Borel sigma algebra (this matters later).
2 Definition of standard Borel space
A measurable space \((X,\mathcal{B})\) is a standard Borel space if there exists a Polish topology \(\tau\) on \(X\) such that \[ \mathcal{B} = \mathcal{B}(X,\tau), \] the Borel sigma algebra generated by \(\tau\). An equivalent viewpoint is \((X,\mathcal{B})\) is Borel-isomorphic to a Borel subset of a Polish space. This is the core object in descriptive set theory and in much of modern probability.
A probability space \((X,\mathcal{B},\mu)\) is called a standard Borel probability space if \((X,\mathcal{B})\) is standard Borel and \(\mu\) is a probability measure on \(\mathcal{B}\). Sometimes people say “standard probability space” with the same intent (often modulo null sets). Remember:
- “standard Borel” refers to the measurable structure,
- “Wiener measure”, “Poisson law”, etc. are extra structure (a measure on that space).
3 Kuratowski’s Isomorphism Theorem
Let \((X,\mathcal{B}_X)\) and \((Y,\mathcal{B}_Y)\) be standard Borel spaces. Then:
- If \(X\) and \(Y\) are countable, they are Borel-isomorphic if and only if \(|X|=|Y|\).
- If \(X\) and \(Y\) are uncountable, then they are Borel-isomorphic.
(Uncountable means cardinality \(>\mathbb N\), not cardinality equal to \(\mathcal c\) so part 2 is much stronger than part 1.) So, among standard Borel spaces: countable ones are classified by cardinality, and there is exactly one uncountable isomorphism type. Equivalently every uncountable standard Borel space is Borel-isomorphic to \(([0,1],\mathcal{B}([0,1]))\), and also to \((2^{\mathbb{N}},\mathcal{B})\) and also to \((\mathbb{N}^{\mathbb{N}},\mathcal{B})\). These spaces are not generally homeomorphic, but they are Borel-isomorphic.
It is tempting to say “isomorphism depends only on cardinality”, but that is false for arbitrary measurable spaces. On a set \(S\) of cardinality continuum, you can put:
- the power set sigma algebra \(\mathcal{P}(S)\), or
- a countably generated Borel sigma algebra (e.g. identify \(S\) with \([0,1]\) and use Borel sets).
These measurable spaces are not isomorphic because their sigma algebras have different cardinalities: \[ |\mathcal{P}(S)| = 2^{2^{\aleph_0}}, \qquad |\mathcal{B}([0,1])| = 2^{\aleph_0}. \] Kuratowski’s theorem is special to the class of standard Borel spaces.
3.1 Sketch proof
- Any standard Borel space is Borel-isomorphic to a Borel subset of a Polish space, and one can code it inside \(2^{\mathbb{N}}\) (Cantor space) or \(\mathbb{N}^{\mathbb{N}}\) (Baire space).
- If the space is uncountable, then the corresponding Borel subset is uncountable.
- A classical theorem says every uncountable Borel subset of a Polish space contains a perfect subset. In particular, it contains a copy of Cantor space \(2^{\mathbb{N}}\) (at least as a closed subset, hence Borel).
- Conversely, the Borel subset itself already sits inside a canonical Polish space.
- Using a measurable Schroder-Bernstein theorem (Borel injections both ways imply Borel isomorphism), conclude the space is Borel-isomorphic to \(2^{\mathbb{N}}\).
This is one of the main reasons standard Borel spaces are so useful: measurable classification becomes very simple.
4 Example 1: Borel on \([0,1]\)
This is the canonical uncountable standard Borel space. Take \(X=[0,1]\) with the usual Euclidean topology. Then:
- \([0,1]\) is compact metric, hence Polish.
- \(\mathcal{B}([0,1])\) is its Borel sigma algebra.
So \(([0,1],\mathcal{B}([0,1]))\) is a standard Borel space.
On this same measurable space you can place many different probability measures:
- Lebesgue measure \(\lambda\),
- a Dirac mass \(\delta_{1/2}\),
- a Bernoulli two-point law,
- a Cantor distribution,
- any probability law on \([0,1]\).
This is a useful reminder: the Borel sigma algebra is the measurable skeleton; the measure is extra data.
5 Example 2: Wiener space \(C(\mathbb{R}_+,\mathbb{R})\), construction, and Wiener measure
There are two related “Wiener spaces” in common use:
- the path space \(C(\mathbb{R}_+,\mathbb{R})\) equipped with its Borel sigma algebra and Wiener measure,
- the classical Banach space \(C_0([0,T])\) (continuous paths starting at 0 on a finite horizon) with the Wiener law.
Let \[ C(\mathbb{R}_+,\mathbb{R}) := \set{\omega:\mathbb{R}_+\to\mathbb{R} \mid \omega \text{ continuous}}. \] A standard metric inducing uniform convergence on compact intervals is \[ d(\omega,\eta) := \sum_{n=1}^\infty 2^{-n}\left(1 \wedge \sup_{0\le t\le n} |\omega(t)-\eta(t)|\right). \] With this topology: \(C(\mathbb{R}_+,\mathbb{R})\) is Polish, and so its Borel sigma algebra makes it a standard Borel space. The canonical coordinate process is \[ X_t(\omega):=\omega(t). \] Wiener measure \(W\) is the law of Brownian motion on this path space. It is characterized by:
- \(X_0=0\) almost surely,
- independent increments,
- for \(0\le s<t\), \(X_t-X_s \sim N(0,t-s)\),
- continuous sample paths almost surely.
Standard construction:
- Define consistent finite-dimensional distributions for \((X_{t_1},\dots,X_{t_n})\) using centered Gaussian increments.
- Use Kolmogorov’s extension theorem to get a probability measure on \(\mathbb{R}^{\mathbb{R}_+}\) (the product space of all functions).
- Use Kolmogorov continuity theorem (or equivalent regularity results) to show the process has a continuous modification.
- Push the law to \(C(\mathbb{R}_+,\mathbb{R})\).
The resulting probability measure on \(\mathcal{B}(C(\mathbb{R}_+,\mathbb{R}))\) is Wiener measure \(W\). Basic properties of Wiener measure:
- \(W\) is a probability measure on a standard Borel space.
- \(W\) is atomless.
- The canonical process under \(W\) is Brownian motion (analog of \(U(\omega)=\omega\) is uniform for \([0,1]\)).
- The natural filtration (completed and right-continuous) is the Brownian filtration.
\(C(\mathbb{R}_+,\mathbb{R})\) and its Borel sigma algebra are purely measurable/topological objects; Wiener measure \(W\) is one particular probability measure on that measurable space.
6 Example 3: Poisson jump process, path space construction, and associated measure
Here the natural path space is the cadlag path space, usually denoted \(D\). Let \[ D(\mathbb{R}_+,\mathbb{R}) := \{\omega:\mathbb{R}_+\to\mathbb{R}\mid \omega \text{ is cadlag}\}. \] “cadlag” means right-continuous with left limits. Equip \(D(\mathbb{R}_+,\mathbb{R})\) with the Skorokhod \(J1\) topology. This topology is designed to handle jumps while allowing small time changes. It is a classical fact that \((D(\mathbb{R}_+,\mathbb{R}),J1)\) is Polish. Hence its Borel sigma algebra is a standard Borel sigma algebra. Again the canonical process is \[ X_t(\omega):=\omega(t). \]
A (homogeneous) Poisson process \((N_t)_{t\ge 0}\) with rate \(\lambda>0\) is characterized by:
- \(N_0=0\),
- independent increments,
- stationary increments,
- \(N_t-N_s \sim \mathrm{Poisson}(\lambda(t-s))\) for \(0\le s<t\),
- sample paths are cadlag step functions with jumps of size \(1\).
So it naturally lives in \(D(\mathbb{R}_+,\mathbb{R})\), and in fact in the subset of integer-valued, nondecreasing step paths. There are several equivalent constructions. Via exponential waiting times: let \((E_n)_{n\ge 1}\) be iid exponential\((\lambda)\) random variables and define jump times \[ T_n := E_1+\cdots+E_n. \] Then define \[ N_t := \sum_{n\ge 1} 1_{\{T_n\le t\}}. \] This gives a Poisson process, and its law is a probability measure on \(D(\mathbb{R}_+,\mathbb{R})\).
As with Brownian motion, one can specify consistent finite-dimensional distributions and use extension theorems, then verify path regularity.
Denote the law by \(P^\mathrm{Pois}\) (or similar notation). This is the pushforward of the underlying probability measure under the path map. This law is a probability measure on the standard Borel space \[ (D(\mathbb{R}_+,\mathbb{R}),\mathcal{B}(D)). \] Atom issue: on a finite horizon \([0,T]\), the zero path has positive probability: \[ \mathsf{P}(N_t=0 \text{ for all } t\le T)=e^{-\lambda T}>0. \] So the finite-horizon path law has an atom. On the infinite horizon \(\mathbb{R}_+\), the event “no jumps ever” has probability \(0\), and the law is atomless (for the standard homogeneous Poisson process). This matters for measure-space isomorphism to Lebesgue space.
8 Definitions: adapted, stopping times, predictable, announceable, inaccessible, etc.
Now we move to filtered probability spaces. Let \[ (\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathsf{P}) \] be a filtered probability space. Usually we assume the usual conditions (right-continuity and completeness), especially in martingale theory.
A process \((X_t)_{t\ge 0}\) is adapted if for every \(t\), \[ X_t \text{ is } \mathcal{F}_t\text{-measurable}. \] Interpretation: at time \(t\), the value \(X_t\) is known from information available by time \(t\).
A random time \(\tau:\Omega\to[0,\infty]\) is a stopping time if for every \(t\ge 0\), \[ \{\tau\le t\}\in\mathcal{F}_t. \] Interpretation: by time \(t\), you can decide whether the event has happened yet. Equivalent versions (under right-continuity, with minor caveats) use \(\{\tau<t\}\).
Examples:
- Hitting times of closed/open sets for continuous adapted processes (with standard regularity assumptions),
- First jump time of a Poisson process.
The optional and predictable sigma algebras on \(\Omega\times\mathbb{R}_+\) used to formalize process regularity.
- The optional sigma algebra is generated by adapted cadlag processes.
- The predictable sigma algebra is generated by adapted left-continuous processes (or by simple rectangles of the form \(A\times(s,t]\) with \(A\in\mathcal{F}_s\)).
Heuristic: optional = information available “at time \(t\)”, predictable = information available “just before time \(t\)”.
A stopping time \(\tau\) is predictable if there exists a sequence of stopping times \((\tau_n)\) such that:
- \(\tau_n < \tau\) on \(\{\tau>0\}\),
- \(\tau_n \uparrow \tau\) pointwise.
Such a sequence \((\tau_n)\) is called an announcing sequence for \(\tau\). This matches the idea “the time can be seen coming from earlier times.”
Examples (continuous-path setting): many hitting times of continuous processes are predictable (e.g. first hitting a level can often be announced by first hitting nearby levels).
A stopping time \(\tau\) is accessible if there exists a sequence of predictable stopping times \((\sigma_n)\) such that \[ \mathsf{P}\left(\tau<\infty,\ \tau\ne \sigma_n \text{ for all } n\right)=0. \] Interpretation: \(\tau\) may not itself be predictable, but it can be exhausted by countably many predictable times. Accessible is weaker than predictable.
A stopping time \(\tau\) is totally inaccessible if for every predictable stopping time \(\sigma\), \[ \mathsf{P}(\tau=\sigma<\infty)=0. \] Interpretation: the time cannot be “caught” by any predictable schedule. It is a pure surprise time. Canonical example: the first jump time \(T_1\) of a Poisson process is totally inaccessible.
8.1 Predictable vs inaccessible in jump theory
For cadlag semimartingales, jumps are often analyzed through their jump measure. Jump times split (roughly) into:
- predictable (announced) jump times,
- totally inaccessible jump times.
This decomposition is a central theme in the general theory of processes.
Brownian motion has no jumps, so this whole jump-time structure becomes trivial there.
9 Results about martingales
9.1 Continuous martingales
A martingale (or local martingale) \(M\) is continuous if almost all sample paths \(t\mapsto M_t(\omega)\) are continuous.
A filtration \((\mathcal{F}_t)\) is often called continuous if every local martingale adapted to it is continuous.
This is true for the Brownian filtration (after usual augmentation).
9.2 Brownian filtration: all local martingales are continuous
Let \((B_t)\) be Brownian motion and let \((\mathcal{F}_t^B)\) be its completed, right-continuous natural filtration.
Main result (martingale representation + continuity):
- Every local martingale \(M\) adapted to \((\mathcal{F}_t^B)\) admits a representation \[ M_t = M_0 + \int_0^t H_s\, dB_s \] for some predictable integrand \(H\) (under suitable integrability/localization conditions).
Since stochastic integrals with respect to continuous semimartingales are continuous, \(M\) is continuous.
This is why people say “the Brownian filtration is continuous.”
This is not about the Borel sigma algebra on path space. It is about the filtration generated by the canonical process under Wiener measure.
9.3 Poisson filtration: jump martingales and totally inaccessible jump times
Let \((N_t)\) be a Poisson process with rate \(\lambda\), and let \((\mathcal{F}_t^N)\) be its natural filtration (usual augmentation). Then: \[ M_t := N_t - \lambda t \] is a martingale, and it has jumps of size \(1\). So the Poisson filtration is not continuous. Indeed, it is the prototypical purely discontinuous filtration (or at least contains a purely discontinuous martingale component). The first jump time \[ T_1 := \inf\{t>0: N_t\ge 1\} \] is totally inaccessible. This expresses the “surprise jump” nature of Poisson noise.
9.4 Comparison table
| Item | Brownian / Wiener | Poisson |
|---|---|---|
| Path space | \(C(\mathbb{R}_+,\mathbb{R})\) | \(D(\mathbb{R}_+,\mathbb{R})\) |
| Paths | continuous | cadlag jump paths |
| Canonical filtration | Brownian filtration | Poisson filtration |
| Natural martingales | continuous | discontinuous (e.g. \(N_t-\lambda t\)) |
| Jump times | none (for the canonical process) | totally inaccessible |
| Typical hitting times | often predictable (announceable) | “Surprise” events |
Remarks.
- Approximation of measurable sets by nicer sets (open/closed, inner/outer regularity, etc.) is a measure/topology fact.
- Predictability and inaccessible times are filtration/time facts.
- Continuity of martingales in the Wiener setting comes from the Brownian filtration and martingale representation, not from Borel regularity of the path space.
- Predictable means “can be approximated from the left in time,”
- Totally inaccessible means “cannot be approached using only prior information.”
10 Hierarchy
Layer A: standard Borel structure
- \([0,1]\) with Borel sigma algebra,
- \(C(\mathbb{R}_+,\mathbb{R})\) with Borel sigma algebra,
- \(D(\mathbb{R}_+,\mathbb{R})\) with Borel sigma algebra.
- all uncountable standard Borel spaces are Borel-isomorphic.
Layer B: probability measure on that space - Lebesgue on \([0,1]\), - Wiener measure on \(C\), - Poisson law on \(D\). - a generic Borel isomorphism does not preserve these measures. - but atomless standard probability spaces are measure-isomorphic mod \(0\) to Lebesgue space.
Layer C: filtration and process structure - Brownian natural filtration (continuous), supports only continuous local martingales. - Poisson natural filtration (jump/discontinuous), has jump martingales and totally inaccessible jump times.
11 Process on Omega vs canonical path space
There are always two equivalent ways to describe a stochastic process.
As a process on an abstract probability space: start with \[ (\Omega,\mathcal{F},\mathsf{P}) \] and an adapted process \[ X=(X_t)_{t\ge 0}. \] Each outcome \(\omega\in\Omega\) produces one sample path \[ t\mapsto X_t(\omega). \]
Or as the canonical process on a path space: choose a path space \(E\) (for example \(C(\mathbb{R}_+,\mathbb{R})\) or \(D(\mathbb{R}_+,\mathbb{R})\)), put a probability law \(\mu\) on it, and define the coordinate process \[ \xi_t(\eta):=\eta(t), \qquad \eta\in E. \] Then the randomness is “the random path itself.” These are the same after pushforward by the path map. The bridge between the two is the path map. Given a process \(X\) on \((\Omega,\mathcal{F},\mathsf{P})\), define the path map \[ \Phi:\Omega\to E, \qquad \Phi(\omega)=\big(t\mapsto X_t(\omega)\big). \]
Here \(E\) is the right path space for the sample-path regularity:
- continuous paths -> \(E=C(\mathbb{R}_+,\mathbb{R}^d)\),
- cadlag paths -> \(E=D(\mathbb{R}_+,\mathbb{R}^d)\).
Then the law of the whole process is the pushforward \[ \mu := \Phi_\#\mathsf{P}. \] This is the path-space law of \(X\). Here \(\mu\) is not just the law of one random variable \(X_t\), it is the law of the entire trajectory.
12 Canonical process on path space
On \(E\), define the canonical (coordinate) process \[ \xi_t(\eta):=\eta(t). \] Under the measure \(\mu=\Phi_\#\mathsf{P}\), the process \(\xi\) has the same finite-dimensional laws as \(X\) (and in fact the same path law by construction). So:
- \(X\) on \((\Omega,\mathcal{F},\mathsf{P})\),
- \(\xi\) on \((E,\mathcal{B}(E),\mu)\)
are two realizations of the same stochastic process law.
This is why people often move to canonical path space – it removes irrelevant details of the original \(\Omega\).
13 Natural filtration
Given a process \(X\) on \(\Omega\), its natural filtration is \[ \mathcal{F}_t^X := \sigma(X_s: 0\le s\le t). \]
Given the canonical process \(\xi\) on path space, its natural filtration is \[ \mathcal{G}_t := \sigma(\xi_s: 0\le s\le t). \]
These correspond under the path map: \[ \Phi^{-1}(A)\in \mathcal{F}_t^X \quad \text{for } A\in\mathcal{G}_t. \] In practice one often uses the usual augmentation (complete + right-continuous version), but the raw natural filtration is the starting point.
14 The “law of the process”
There are three related notions that are easy to confuse.
- One-time marginal law. For fixed \(t\), \[ \mathcal{L}(X_t)=\mathsf{P}\circ X_t^{-1}. \]
- finite-dimensional distributions. For \(0\le t_1<\cdots<t_n\), \[ \mathcal{L}(X_{t_1},\dots,X_{t_n}). \]
- Path law (strongest) \[ \mathcal{L}(X)=\Phi_\#\mathsf{P}, \] a measure on path space \(E\).
For continuous/cadlag processes, (c) is the cleanest object.
14.1 Wiener case
A Brownian motion \(B=(B_t)\) lives on some \((\Omega,\mathcal{F},\mathsf{P})\).
Path map: \[ \Phi_B(\omega)=\big(t\mapsto B_t(\omega)\big)\in C(\mathbb{R}_+,\mathbb{R}). \]
Pushforward: \[ W := (\Phi_B)_\#\mathsf{P}. \] This \(W\) is Wiener measure. On \[ (E,\mu)=\big(C(\mathbb{R}_+,\mathbb{R}),W\big), \] the coordinate process \[ \xi_t(\eta)=\eta(t) \] is Brownian motion. So “Wiener measure” is exactly “the law that makes the coordinate process Brownian.”
14.2 Poisson case
A Poisson process \(N=(N_t)\) lives on some \((\Omega,\mathcal{F},\mathsf{P})\).
Path map: \[ \Phi_N(\omega)=\big(t\mapsto N_t(\omega)\big)\in D(\mathbb{R}_+,\mathbb{R}). \]
Pushforward: \[ P^{\mathrm{Pois}} := (\Phi_N)_\#\mathsf{P}. \]
This is the Poisson path law. On \[ (E,\mu)=\big(D(\mathbb{R}_+,\mathbb{R}),P^{\mathrm{Pois}}\big), \] the coordinate process \[ \xi_t(\eta)=\eta(t) \] is a Poisson process.
Again, the named measure is the law of the coordinate process.
15 References
- Kallenberg, Foundations of Modern Probability
- Revuz and Yor, Continuous Martingales and Brownian Motion
- Rogers and Williams, Diffusions, Markov Processes and Martingales
- Dellacherie and Meyer (or He, Wang, and Yan) for predictable/optional sigma algebras and stopping times
- Cohn (Measure Theory) or Kechris (Classical Descriptive Set Theory) for standard Borel spaces and Borel isomorphism results
7 Comments
The three spaces
are all uncountable standard Borel spaces. Therefore, by Kuratowski’s theorem, they are all Borel-isomorphic, meaning there exists a bijection between any pair whose forward and inverse maps are Borel-measurable. This is a statement about measurable structure only. It ignores the probability measures.
Now add the “distinguished” measures:
A generic Borel isomorphism does not preserve these measures.
If \(f:X\to Y\) is a Borel isomorphism and \(\mu\) is a measure on \(X\), then \(f\) produces the pushforward measure \(f_\#\mu\) on \(Y\), but usually: \[ f_\#\mu \ne \nu \] for the particular “named” measure \(\nu\) you care about.
The isomorphism theorem for atomless standard probability spaces, or the Rokhlin (Rohlin) isomorphism theorem says that any atomless standard probability space is isomorphic mod null sets to \(([0,1],\mathcal{B},\lambda)\).
This is a measure-preserving statement (mod \(0\)), not just a Borel-isomorphism statement. As a consequence:
A Borel isomorphism preserves:
A Borel isomorphism does not preserve: