Brownian motion

Assume the Axiom of Choice The Axiom of Choice. A real process $B=(B_t{)}_{t\ge 0}$ is a standard Brownian motion if:

1. $B_0=0$ almost surely;
2. for every finite list $0=t_0<t_1<\cdots <t_n$, the increments $B_{t_j}-B_{t_{j-1}}$ are mutually independent and have laws $N(0,t_j-t_{j-1})$; and
3. there is one measurable event $A$ with $P(A)=1$ such that $t\mapsto B_t(\omega )$ is continuous on $[0,\mathrm{\infty })$ for every $\omega \in A$.

By Brownian covariance is equivalent to independent stationary normal increments, the first two clauses are equivalent to saying that $B$ is a centered Gaussian process with covariance $\min (s,t)$. Clause 3 is additional: it cannot be recovered from finite-dimensional distributions alone.

No filtration is part of this definition. In particular, no completed or right-continuous filtration and no Markov or martingale assertion is silently imposed. Modifications and indistinguishability retain the distinct meanings in Process law, modification, and indistinguishability. The cases $n=0$, $t_1=0$, and zero-length increments are respectively vacuous or already covered by $B_0=0$; the increment list itself is strictly increasing.

Choice is declared because the normal-law and Gaussian/increment-equivalence interfaces construct and identify normal laws under AC. The continuity clause selects no path and makes no additional use of choice.

The Axiom of Choice

Brownian covariance is equivalent to independent stationary normal increments

Process law, modification, and indistinguishability

Sousi, Section 6.1 (printed p. 51), gives these three defining clauses. The common full-measure event formulation makes the pathwise quantifier explicit.