The Brownian zero set has no isolated points

Apply [F4] to X and -X on each interval [0,1/m], m>=1. Off a countable union of null events [F5], X has both a positive and a negative value on each such interval. Neither value is at zero. The intermediate value theorem between those two times gives a zero at a positive time <=1/m. Therefore positive zeros accumulate at zero. The same argument applies to the coordinate process under Wiener measure, which is Brownian by [F3].

Fix a nonnegative rational q and define ${\tau }_q=\inf \{s\ge q:X_s=0\}$, with infimum of the empty set infinity. For t<q its sublevel event is empty. For t>=q, compactness and closedness give $\{{\tau }_q\le t\}=\{{\min }_{r\in [q,t]}\mathrm{\mid }{X}_r\mathrm{\mid }=0\}$, which equals the event that the infimum over $(\mathbb{Q}\cap [q,t])\cup \{q,t\}$ is zero. At t=q this is the singleton test X_q=0. Every coordinate used has time <=t, so the event belongs to the raw filtration of X and thus to its usual augmentation [F2]. This proves that tau_q is a stopping time. If tau_q is finite, closedness gives X_{tau_q}=0. No recurrence or finiteness assertion is needed.

Define $G(w)=1_{{\bigcap }_{m\ge 1}{\bigcup }_{k>m}\{{\inf }_{r\in \mathbb{Q}\cap [1\mathrm{/}k,1\mathrm{/}m]}\mathrm{\mid }w(r)\mathrm{\mid }=0\}}$. This is product measurable, since only countably many coordinates and set operations occur. For continuous w the compact infimum is a minimum and equals the rational infimum, so G(w)=1 precisely when positive zeros accumulate at zero. Step 1.1 shows $\mathrm{\Psi }(0)=\int G(w)\mu (dw)=1$. For any nonzero x, continuity of w at zero and w(0)=0 exclude zeros of x+w sufficiently near zero; hence $\mathrm{\Psi }(x)=0$. These assertions hold mu-almost surely even if a realization of Wiener measure contains nonzero-start null paths.

For integers N>=1 put S=tau_q wedge N. It is a bounded stopping time: its sublevel event is {tau_q<=t} for t<N and the whole space for t>=N. By [F3] and step 2.1, $E[G((X_{S+r}{)}_{r\ge 0})\mid {\mathcal{G}}_S]=1_{\{X_S=0\}}$. The event {X_S=0} is in the stopped sigma-algebra by [F3]. Integrating the nonnegative variable $1_{\{X_S=0\}}(1-G((X_{S+r})))$ over the whole space gives zero by the conditional event-integral identity [F6]; since this variable is an indicator, the event it indicates is null. Thus, almost surely on {X_S=0}, positive zeros of the post-S path accumulate at zero. On {tau_q<=N} step 1.2 gives S=tau_q and X_S=0. We conclude that, outside a measurable null event, whenever tau_q<=N the zero tau_q is approached by other zeros from the right.

Take the intersection of the full event in step 1.1 with the full events in step 3.1 over the countable set of nonnegative rationals q and positive integers N. It is a measurable full event by [F5]. On it, every finite tau_q is approached by other zeros from the right, since some N is larger than tau_q.

On that event zero is not isolated by step 1.1. If t>0 were an isolated zero, some delta with 0<delta<t would satisfy $Z\cap (t-\delta ,t+\delta )=\{t\}$. By rational density choose q in (t-delta,t). There are no zeros in [q,t), and t is a zero, so tau_q=t. Step 4.1 supplies other zeros in (t,t+delta), a contradiction. Hence every zero is a limit point of the other zeros. Z is already closed by [F1].

The empty hitting set and tau_q=infinity cause no problem because only bounded S are used, and the conclusion about tau_q is conditional on its finiteness. The time-zero endpoint has its own full event. The only intersections of full events are countable; no rational time is claimed itself to be a positive Brownian zero. On the common agreement event in [F1] the original B and X have identical zero sets, so the original path assertion follows there. AC is inherited as in [F6]. In particular, the proof never places an arbitrary normalization null event in the original Brownian filtration.