= Solution
Let $\mu$ be the given <Brownian motion> law on $C[0,1]$. Choose $\alpha<\beta<1/2$ and an integer $p$ with $p(1/2-\beta)>1$. For the largest adjacent level-$n$ increment $A_n$, <Brownian scaling> and the <Markov inequality> give
$$
\mu(A_n>2^{-n\beta})
\leq\sum_{k=0}^{2^n-1}\frac{\mathbb E|X_{(k+1)2^{-n}}-X_{k2^{-n}}|^p}{2^{-np\beta}}
=c_p\,2^{-n(p(1/2-\beta)-1)},
$$
where $c_p=\mathbb E|\mathcal N(0,1)|^p<\infty$. These probabilities are summable, so the <Borel-Cantelli lemma> gives a finite random $K$ such that $A_n\leq K2^{-n\beta}$ for every $n$, after absorbing the finitely many exceptional levels.
For $s<t$, choose $m$ with $2^{-m}\leq t-s<2^{1-m}$. Their left dyadic approximations at level $m$ differ by at most two grid steps; each finer approximation adds at most one adjacent increment. Continuity and <dyadic increment chaining> therefore give
$$
|X_t-X_s|\leq2\sum_{j\geq m}A_j
\leq\frac{2K}{1-2^{-\beta}}\,2^{-m\beta}
\leq\frac{2K}{1-2^{-\beta}}(t-s)^\beta.
$$
Thus paths are $\beta$-Hölder and hence $\alpha$-Hölder on $[0,1]$. This direct proof supplies the relevant <Brownian Hölder regularity>, also obtainable from the <Kolmogorov continuity theorem>.
The full-measure subset $E_\alpha=C^\alpha[0,1]$ is measurable in $C[0,1]$, because continuity identifies its seminorm with a countable supremum:
$$
\|w\|_\alpha
=\sup_{s<t,\ s,t\in\mathbb Q\cap[0,1]}
\frac{|w(t)-w(s)|}{(t-s)^\alpha}.
$$
Its finiteness event is a countable union of countable intersections of coordinate-measurable events. The sigma-algebra specified on $E_\alpha$ is precisely the trace of the <coordinate sigma-algebra> on $C[0,1]$. Define $\nu(A)=\mu(A)$ for trace-measurable $A\subseteq E_\alpha$; equivalently restrict the Brownian law to its full-measure subset. Since $\mu(E_\alpha)=1$, no renormalization is needed, and all finite-dimensional coordinate laws are unchanged. \b[This is the required <probability measure>]:
$$
\boxed{\nu(C^\alpha[0,1])=1,\qquad (X_t)_{0\leq t\leq1}\text{ is Brownian under }\nu.}
$$
The construction uses the requested coordinate sigma-algebra and makes no assumption about a separable Hölder-norm topology.
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