LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Brownian motion has a jointly measurable continuous version

Statement

Let $B$ be a standard Brownian motion Brownian motion on a probability space $(Ω, F, P)$ . Then there is a process $\hat{B} = ({\hat{B}}_{t})_{t \geq 0}$ with the following properties.

$\hat{B}$ is indistinguishable from $B$ , and ${\hat{B}}_{0} = 0$ on all of $Ω$ .
Every path $t \mapsto {\hat{B}}_{t} (ω)$ is continuous on $[0, \infty)$ , so $ω \mapsto \hat{B} (ω)$ is a map into the path space $C ([0, \infty), R)$ of Uniform-on-compacts metric on continuous path space.
That map is Borel measurable for the uniform-on-compacts Borel $σ$ -algebra, and $(t, ω) \mapsto {\hat{B}}_{t} (ω)$ is measurable for $B ([0, \infty)) \otimes F$ .

The version is obtained by keeping the original process on one probability-one event and replacing the whole path by the zero path outside it; no distribution of $B$ is altered and no path is selected by any choice principle.

Facts & Assumptions

Given: AC and a standard Brownian motion $B$ on $(Ω, F, P)$ .

[F1]

There is a measurable event $A$ with $P (A) = 1$ on which every path $t \mapsto B_{t} (ω)$ is continuous, and $B_{0} = 0$ almost surely. Brownian motion

[F2]

On $C ([0, \infty), R)$ the uniform-on-compacts metric induces the topology of uniform convergence on compact subsets of $[0, \infty)$ . Uniform-on-compacts metric on continuous path space

[F3]

The Borel $σ$ -algebra of $C ([0, \infty), R)$ is generated by the coordinate maps $π_{t} (f) = f (t)$ , $t \geq 0$ . Borel sigma-algebra of continuous path space is generated by coordinates

[F4]

The evaluation map $e (f, t) : = f (t)$ on $C ([0, \infty), R) \times [0, \infty)$ is defined for every pair, and it is continuous when the path space carries the uniform-on-compacts topology. The evaluation map $e : C (X, Y) \times X \to Y$ , $e (f, x) = f (x)$

[F5]

AC is the ambient assumption of the Brownian interfaces. The Axiom of Choice

Proof

technique · direct

1.1

Put $A^{'} : = A \cap {B_{0} = 0}$ , a measurable event with $P (A^{'}) = 1$ by [F1], and define ${\hat{B}}_{t} (ω) : = B_{t} (ω)$ for $ω \in A^{'}$ and ${\hat{B}}_{t} (ω) : = 0$ for $ω \notin A^{'}$ ; then $\hat{B}$ is indistinguishable from $B$ because it differs from $B$ only on the null set $(A^{'})^{c}$ , and ${\hat{B}}_{0} (ω) = 0$ for every $ω$ .

givenF1

1.2

The evaluation map of [F4] is continuous: if $f_{m} \to f$ in the metric of [F2] and $t_{m} \to t$ in $[0, \infty)$ , choose $N$ with $t_{m} \leq N$ for all $m$ and $∣ f (s) - f (t) ∣ < ε / 2$ for $∣ s - t ∣ < η$ , which is possible because the continuous $f$ is continuous at $t$ ; convergence in the metric of [F2] gives $\sup_{0 \leq s \leq N} ∣ f_{m} (s) - f (s) ∣ < ε / 2$ for all large $m$ , and then $∣ f_{m} (t_{m}) - f (t) ∣ \leq \sup_{0 \leq s \leq N} ∣ f_{m} (s) - f (s) ∣ + ∣ f (t_{m}) - f (t) ∣ < ε$ for all large $m$ .

F2F4

2.1

Every path of $\hat{B}$ is continuous: for $ω \in A^{'}$ it is the Brownian path, continuous by [F1], and for $ω \notin A^{'}$ it is the zero path.

step 1.1F1

2.2

Each time coordinate of $\hat{B}$ is measurable: ${\hat{B}}_{t} = 1_{A^{'}} B_{t}$ is the product of a measurable indicator with the measurable function $B_{t}$ .

step 1.1

3.1

The map $ω \mapsto \hat{B} (ω)$ is a random element of $C ([0, \infty), R)$ with its Borel $σ$ -algebra: by [F3] that $σ$ -algebra is $σ (π_{t} : t \geq 0)$ , and for every $t$ and every Borel $E \subseteq R$ one has ${ω : π_{t} (\hat{B} (ω)) \in E} = {{\hat{B}}_{t} \in E}$ , measurable by [step 2.2]; the family of Borel sets with measurable preimage is a $σ$ -algebra containing the generating sets $π_{t}^{- 1} (E)$ , hence all of $B (C)$ .

step 2.1step 2.2F3

4.1

The pair map $(t, ω) \mapsto (\hat{B} (ω), t)$ is measurable from $B ([0, \infty)) \otimes F$ into the product of the Borel $σ$ -algebras, because its first component is measurable by [step 3.1] and its second is a coordinate projection; composing with the continuous, hence Borel measurable, evaluation map of [step 1.2] shows that $(t, ω) \mapsto {\hat{B}}_{t} (ω)$ is $B ([0, \infty)) \otimes F$ -measurable.

step 3.1step 1.2

5.1

The boundary cases are covered: the normalization changes the process only on the null set $(A^{'})^{c}$ , so indistinguishability and every almost-sure statement are preserved; the initial value is $0$ on all outcomes as required; the case $t = 0$ enters the measurability statements through the coordinate ${\hat{B}}_{0}$ , which is identically zero; and AC enters only through [F5] as the ambient assumption of the Brownian definition.

step 1.1step 2.1step 4.1F5given∎

Source notes

Durrett, Section 7.1, fixes a continuous version of Brownian motion as part of the definition of the process; Sousi's Chapter 6 treats Brownian motion as a random continuous path. The lemma records the two measurability consequences used later on the page: the path map is a random element of the continuous path space, and the evaluation map makes the process jointly measurable, which is what the Tonelli and Fubini arguments for the zero set and the occupation time need.

Depends on

Used by

The Brownian zero set Definition
Exponential martingale Brownian tail bound Example
Ito formula for Brownian powers Example
The Brownian p-variation threshold Example
Brownian step-potential resolvent at zero Lemma
The Brownian zero set has Lebesgue measure zero Lemma
Brownian positive occupation time has the arcsine law Theorem
Dynkin formula for bounded Brownian stopping Theorem

Dependency tree · two levels

38 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Rick Durrett, Probability: Theory and Examples, fifth edition, Section 7.1 (Brownian path normalization) (standard reference, not scraped)
Perla Sousi, Advanced Probability, Chapter 6 (Brownian motion as a random continuous path) (standard reference, not scraped)