Can we show that a countable "linear combination" of i.i.d standard normal distributed RVs have mean equal to zero?
Let $(X_k)_{k \in \mathbb{N}} \sim \mathcal{N}(0,1)$ be a sequence of i.i.d. standard normally distributed random variables on some probability space $(\Omega, \Sigma, \mathbb{P})$. Further let $H := L^2(\mathbb{R}_+, dx)$ be equipped with the inner product $\langle f,g \rangle_H = \int_0^{\infty} f(x)g(x) dx$ and let $(b_n)_{n \in \mathbb{N}}$ be an orthonormal Schauder-basis of $H$. Now for $h \in H$ define $$Z_n := \sum_{k = 1}^{n} \langle h, b_k \rangle_H X_k. $$ By linearity of $\mathbb{E}$ we can compute the expected value of $Z_n$: $$\forall n : \quad \mathbb{E}[Z_n] = \mathbb{E}\left[\sum_{k = 1}^{n} \langle h, b_k \rangle_H X_k\right] = \sum_{k = 1}^{n} \langle h, b_k \rangle_H \mathbb{E}\left[X_k\right] = 0,$$ where we used the fact that $\mathbb{E}[X_k] = \mu = 0$ for all $k \in \mathbb{N}$. Can you help me to solve this for $n \to \infty$? $$ \mathbb{E}[ \lim_{n \to \infty} Z_n] = \mathbb{E}\left[\lim_{n \to \infty} \sum_{k = 1}^{n} \langle h, b_k \rangle_H X_k\right] \stackrel{?}{=} 0,$$
I want to use it for another proof and know, that it has to be true, but I can't think of a proof. Maybe it's possible to use Beppo Levi's monotone convergence theorem for Lebesgue integral? But it's not clear to me, that the sequence is pointwise non-decreasing. Another attempt, would be to use Kolmogorov's extension theorem maybe, but there I'm also lost. Help would be appreciated.