The Brownian zero set

Assume the Axiom of Choice and let $B$ be a standard Brownian motion. Replace $B$, as permitted by Brownian motion has a jointly measurable continuous version, by the indistinguishable version $\widehat{B}$ whose every path is continuous and whose evaluation $(t,\omega )\mapsto {\widehat{B}}_t(\omega )$ is jointly measurable. The Brownian zero set is $Z:=Z(\widehat{B}):=\{t\ge 0:{\widehat{B}}_t=0\},$ and for a horizon $T>0$ one writes $Z_T:=Z\cap [0,T]$.

The following are part of the definition and are used later in this form.

1. Pathwise closedness. For every outcome the set $Z$ is closed in $[0,\mathrm{\infty })$ and nonempty: it is the preimage of the closed set $\{0\}$ under the continuous path, and ${\widehat{B}}_0=0$ for every outcome. Hence $Z_T$ is compact for every $T<\mathrm{\infty }$.
2. Version independence. Replacing $\widehat{B}$ by the original $B$ changes $Z$ only on a $P$-null set: $B$ and $\widehat{B}$ are indistinguishable. Every almost-sure assertion about $Z$ proved below is therefore an almost-sure assertion about the zero set of $B$, and no statement below quantifies over versions.
3. Measurability of the section integrals. Joint measurability makes $(t,\omega )\mapsto 1_{\{{\widehat{B}}_t=0\}}$ measurable for $\mathcal{B}([0,\mathrm{\infty }))\otimes \mathcal{F}$, so for each $T$ the section integral $\omega \mapsto \lambda (Z_T(\omega ))={\int }_0^T{1}_{\{{\widehat{B}}_t(\omega )=0\}}dt$ is a measurable function of $\omega$, and the Tonelli identity for the product measure $dt\otimes P$ applies to it.
4. Endpoint conventions. The point $t=0$ belongs to $Z$ for every outcome, the singleton $\{0\}$ has Lebesgue measure zero, and a horizon $T$ may be replaced by any larger horizon since $Z_T\subseteq Z_{T^{\mathrm{\prime }}}$ for $T\le T^{\mathrm{\prime }}$.

No further structure is assigned: in particular $Z$ is not asserted to be perfect, uncountable or of measure zero by this definition; those are statements proved separately on this page.