= Solution
For every $w\in C^\alpha[0,1]$, its finite <Hölder seminorm> gives the deterministic bound
$$
\sum_{k=0}^{2^n-1}\bigl(w((k+1)2^{-n})-w(k2^{-n})\bigr)^2
\leq2^n\|w\|_\alpha^2\,2^{-2n\alpha}
=\|w\|_\alpha^2\,2^{n(1-2\alpha)}.
$$
When $\alpha>1/2$, the right-hand side tends to zero. Thus under any <probability measure> on the stated <coordinate sigma-algebra>, these measurable finite-coordinate sums converge to zero on every path, and hence almost surely.
If the coordinate process were a <Brownian motion>, part (b) would make the same sums converge to one in $L^2$, and hence in probability. <Almost sure convergence> to zero also implies <convergence in probability> to zero. Uniqueness of a limit in probability is a contradiction. Therefore \b[no such Brownian measure exists]:
$$
\boxed{\alpha>\tfrac12\quad\Longrightarrow\quad\text{no Brownian coordinate law on }C^\alpha[0,1].}
$$
This is the <quadratic variation obstruction to Hölder regularity>. It does not address the critical exponent $\alpha=1/2$, where the displayed deterministic bound does not tend to zero.
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