One-dimensional Brownian motion is recurrent

Fix $c\in \mathbb{R}$ and define on continuous path space ${\varphi }_c(v)=1_{\{\mathrm{\exists }t\ge 0:v(t)=c\}}$. This functional is Borel: its one-set is ${\bigcup }_{m\ge 1}\{v:{\min }_{0\le t\le m}\mathrm{\mid }v(t)-c\mathrm{\mid }=0\}$, and each displayed minimum is continuous for uniform convergence on $[0,m]$ (changing the path by at most $\epsilon$ changes the minimum by at most $\epsilon$). For every deterministic $x\in \mathbb{R}$, [F1] applied under Wiener measure to the level $c-x$ gives $\int {\varphi }_c(x+w)\mu (dw)=1$.

Fix $n\ge 1$ and let $A_{c,n}:=\{\mathrm{\exists }t\ge n:B_t=c\}$. Because the chosen representative is everywhere continuous, $1_{A_{c,n}}={\varphi }_c((B_{n+t}{)}_{t\ge 0})$ pointwise. Applying [F2] and step 1.1 at time $n$ therefore gives $P(A_{c,n}\mid {\mathcal{F}}_n)=1$ almost surely; by [F3] this forces $P(A_{c,n})=1$. This conditions a fixed Borel future-path event and evaluates its kernel at the known state $B_n$; it does not apply [F1] directly to a random level.

For fixed $c$ the events $A_{c,1}\supseteq A_{c,2}\supseteq \cdots$ all have probability one, so $A_c:={\bigcap }_{n\ge 1}{A}_{c,n}$ has probability one by [F4], and on $A_c$ the path visits the level $c$ at arbitrarily large times.

The intersection $A:={\bigcap }_{c\in \mathbb{Q}}{A}_c$ over the countable set of rationals again has probability one by [F4]; on $A$, for every rational $c$ and every time bound the path visits $c$ at some larger time.

Let $I\subseteq \mathbb{R}$ be a nonempty open interval. By [F5] choose a rational $c\in I$. On the probability-one event $A$ of step 4.1 the path visits $c$, hence enters $I$, at arbitrarily large times. Since every nonempty open interval arises in this way and $A$ does not depend on $I$, almost surely the set $\{t:B_t\in I\}$ is unbounded for every nonempty open interval $I$.

The equivalent formulation follows: for a real point $y$ and $\epsilon >0$, the interval $(y-\epsilon ,y+\epsilon )$ is nonempty and open, so it is visited at arbitrarily large times almost surely. The case of the empty interval is excluded, singleton intervals are not claimed as infinitely visited except through the containing open intervals, and the conclusion is about the unboundedness of the visit set, not about any integrability of a hitting time; the first visit of a fixed level is the almost-sure finiteness proved in [F1]. AC is used only through [F6].