Brownian paths have infinite total variation

For $Z\sim N(0,1)$ one has $E\mathrm{\mid }Z\mathrm{\mid }=\sqrt{2\mathrm{/}\pi }$ and $\mathrm{Var}(\mathrm{\mid }Z\mathrm{\mid })=1-2\mathrm{/}\pi$: by [F2] and [F3], $E\mathrm{\mid }Z\mathrm{\mid }=2{\int }_0^{\mathrm{\infty }}x\phi (x)dx=2(2\pi {)}^{-1\mathrm{/}2}{\lim }_{R\to \mathrm{\infty }}(1-e^{-R^2\mathrm{/}2})=(2\mathrm{/}\pi {)}^{1\mathrm{/}2}$, while $E\mathrm{\mid }Z{\mathrm{\mid }}^2=E{Z}^2=1$ by [F9] with $m=1$, so [F10] gives variance $1-(2\mathrm{/}\pi )$. The compact substitution integrals in [F3] are transferred to Lebesgue integrals before the nonnegative monotone limit; expectation and density identities in [F3] justify the displayed Gaussian integral.

Fix $n\ge 1$, put $h=(b-a)\mathrm{/}{2}^n$ and $S_n:={\sum }_{k=1}^{2^n}\mathrm{\mid }{B}_{a+kh}-B_{a+(k-1)h}\mathrm{\mid }$; by [F1] and [F2] the $2^n$ increments are independent with the law of $\sqrt{h}Z$, so $E{S}_n=2^n\sqrt{2h\mathrm{/}\pi }=\sqrt{2(b-a)\mathrm{/}\pi }\cdot 2^{n\mathrm{/}2}\to \mathrm{\infty }$ and $\mathrm{Var}(S_n)=2^{n}h(1-2\mathrm{/}\pi )=(b-a)(1-2\mathrm{/}\pi )$ is constant in $n$. Indeed every absolute increment is square-integrable by step 1.1; [F10] applied to the absolute-value Borel functions on any two distinct increments gives zero covariance, and finite bilinearity gives the variance sum. If $a>0$, the independent increment list in [F1] is obtained by including $0,a$ in the time grid and discarding its first increment; for $a=0$ no extra interval is needed.

By [F5] applied to $S_n$ with $\lambda =\frac{1}{2}E{S}_n$, $P(S_n\le \frac{1}{2}E{S}_n)\le 4\mathrm{Var}(S_n)\mathrm{/}(E{S}_n{)}^2=\frac{4(b-a)(1-2\mathrm{/}\pi )}{\frac{2(b-a)}{\pi }{2}^n}=\frac{2\pi (1-2\mathrm{/}\pi )}{2^n}$, which is summable in $n$.

By [F6] and [step 2.1], almost surely $S_n>\frac{1}{2}E{S}_n$ for all sufficiently large $n$, and hence $S_n\to \mathrm{\infty }$; since each $S_n=V(B,P_n)$ is the variation of the path over the dyadic partition $P_n$ of $[a,b]$, the variation sums over partitions of $[a,b]$ are almost surely unbounded above.

The argument of steps 1.1-4.1 depends on $0\le a<b$ only through the single number $b-a$, so it applies verbatim to every ordered pair of nonnegative rationals $a<b$; by [F7] the pairs form a countable family. For each pair use the measurable event ${\bigcup }_N{\bigcap }_{n\ge N}\{S_n>\frac{1}{2}E{S}_n\}$ of probability one from step 4.1; intersecting these explicit events gives a measurable probability-one event on which the variation sums of $B$ are unbounded above on every compact interval in $[0,\mathrm{\infty })$ with rational endpoints.

On that event every nondegenerate compact interval $[c,d]\subseteq [0,\mathrm{\infty })$ has unbounded variation sums as well: choose, by [F7], nonnegative rationals $a<b$ with $c<a<b<d$, note that the dyadic partitions of $[a,b]$ extend to partitions of $[c,d]$ by adding the points $c$ and $d$, and that adding points can only increase a variation sum by the triangle inequality, so the sums over partitions of $[c,d]$ dominate the unbounded family for $[a,b]$.

The boundary cases are covered: the interval is required to be nondegenerate, so $a=b$ and the singleton convention are excluded; $2^n\ge 2$ increments are used, so the $n$-sums are genuine variation sums over partitions in the sense of [F4]; the variance bound is uniform in the mesh index for each fixed interval, while the full-measure intersection in step 5.1 is legitimate by countability, not by an interval-independent variance constant. AC includes the Countable Choice required for the compact Riemann/Lebesgue bridge and is inherited through the Brownian and normal-law interfaces in [F8].