 Solution

Enumerate the orthonormal basis consisting of $H_0$ and the Haar wavelets $H_{n,k}$ as $(e_j{)}_{j\ge 1}$, and enumerate the corresponding independent standard normal random variables as $(A_j{)}_{j\ge 1}$. Set $c_j=⟨h,e_j{⟩}_{\mathcal{H}}$. The Parseval identity for a Hilbertian basis givesThe partial sums $T_m={\sum }_{j=1}^m{c}_j{A}_j$ form a martingale for $\sigma (A_1,\dots ,A_m)$, since the next summand is independent of the past and has mean zero. Moreover,The martingale convergence theorem for a martingale bounded in L2 space proves convergence almost surely and convergence in L2. Equivalently one may apply the Kolmogorov convergence theorem to the summable variances $c_j^2$.

A finite linear combination of independent normal random variables is normal. Its characteristic function is $\mathbb{E}{e}^{iu{T}_m}=\exp (-u^2{\sum }_{j\le m}{c}_j^2/2)$. Passing to the limit identifiesFor $h=0$ this is the constant zero random variable. Each fixed $h$ has its own permissible null set; simultaneous pathwise convergence for all $h\in \mathcal{H}$ is unnecessary.