Brownian paths are nowhere locally one-half Hölder

Fix rationals $0\le a<b$ and an integer $C\ge 1$, and put $E_{a,b,C}:={\bigcap }_{r,q\in \mathbb{Q}\cap [a,b]}\{\mathrm{\mid }{B}_r-B_q\mathrm{\mid }\le C\mathrm{\mid }r-q{\mathrm{\mid }}^{1\mathrm{/}2}\}\mathrm{.}$ This is measurable because it is a countable intersection of coordinate events. For every integer $n\ge 1$, with $h=(b-a)\mathrm{/}n$ and grid points $t_k=a+kh$ (all rational), the event $E_{a,b,C}$ is contained in $A_n:=\{\omega :\mathrm{\mid }{B}_{t_k}(\omega )-B_{t_{k-1}}(\omega )\mathrm{\mid }\le C\sqrt{h}$ for $k=1,\dots ,n\}$, because consecutive grid points are rational pairs in $[a,b]$ at distance $h$.

Insert the endpoint 0 before a when a>0; [F1] then applies to the increasing grid starting at zero, and its subfamily of increments on [a,b] is independent. By [F1] and [F2] the increments $B_{t_k}-B_{t_{k-1}}$, $k=1,\dots ,n$, are independent with the law of $\sqrt{h}Z$, so each satisfies $P(\mathrm{\mid }{B}_{t_k}-B_{t_{k-1}}\mathrm{\mid }\le C\sqrt{h})=P(\mathrm{\mid }Z\mathrm{\mid }\le C)=p_C<1$, and independence gives $P(A_n)=p_C^n$; hence $P(E_{a,b,C})\le p_C^n$ for every $n\ge 1$ and therefore $P(E_{a,b,C})=0$. For completeness, the standard normal probability of [C+1,C+2] is at least $e^{-(C+2{)}^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }>0$, proving p_C<1 for the positive integers C used here.

The family of ordered pairs of rationals and of integers is countable, so [step 2.1], countable subadditivity [F5] and [F4] give $P({\bigcup }_{0\le a<b,a,b\in \mathbb{Q}}{\bigcup }_{C\ge 1}{E}_{a,b,C})=0$.

On the complement of that null event there is no nondegenerate interval $I$ with a finite one-half Hölder constant: if $I$ were such an interval with any finite real constant $C$, then by [F3] we could choose rationals $0\le a<b$ with $[a,b]\subseteq I$, and with $C^{\mathrm{\prime }}:=\lceil \max (C,1)\rceil \in \mathbb{N}$ the bound would in particular hold for all rational $s,t\in [a,b]$, that is, $E_{a,b,C^{\mathrm{\prime }}}$ would occur.

The intended cases are covered: the interval is required to be nondegenerate, so the empty and singleton interval cases are excluded; the value $n=1$ in [step 2.1] is the degenerate single-increment case of the estimate and already gives $P(A_1)=p_C<1$; the union over integers $C\ge 1$ covers every finite real constant up to rounding up; the estimates in [step 2.1] hold for every positive integer n and imply nullness without requiring the mesh events to be nested; and AC is used only through [F4] via [F1] and [F2].