Energy efficient approximations of Brownian Sheet

For a random field $B(t_1, \ldots, t_d), t_i \in [0, T_i]$ with a reproducing kernel $H$ and any function $f\in H$ define approximation error as
$$ \mathcal{E}_{\bar T}(f, B) =\int\limits_0^{T_1}\ldots \int\limits_0^{T_d} (f(\bar t) - B(\bar t))^2 d\bar t + \lambda^2 \|f\|_{H}^2. $$
The first term defines proximity of $f$ to $B$ and the second one defines energy efficiency of $f$. Coefficient $\lambda$ allows to balance between these two parts. The best approximation is
$$ f_{\mathrm{opt}} = \underset{f\in H}{\arg\min}\, \mathcal{E}_{\bar T}(f, B). $$
We prove the law of large numbers on convergence of optimal approximation error of Brownian Sheet in $L^2$ and almost surely.