Define $\mathcal{H}_\lambda$ to be the space of functions $f$ holomorphic in $\mathbb{C}^{n−1}$, for which $$\|f\|_{\mathcal{H}_\lambda}^2 \equiv \int_{\mathbb{C}^{n−1}}|f(z)|^2 e^{-4\pi\lambda|z|^2}dm(z) < \infty.$$
I have managed to prove that $\mathcal{H}_\lambda$ is complete, and hence a Hilbert space.
Now we define the space $\mathcal{H}$ of functions $f(z′ , \lambda)$, with $(z′, \lambda) \in \mathbb{C}^{n−1} \times \mathbb{R}^+$, that are jointly measurable, holomorphic in $z′ \in \mathbb{C}^{n−1}$ for almost every $\lambda$, and for which $$\|f\|_{\mathcal{H}}^2 \equiv \int_0^\infty \int_{\mathbb{C}^{n−1}}|f(z',\lambda)|^2 e^{-4\pi\lambda|z'|^2}dm(z') d\lambda < \infty.$$
I want to prove that with this norm the space $\mathcal{H}$ is complete and hence a Hilbert space.
If we could prove the following,
If $\{ f_n \}$ is a Cauchy sequence in $\mathcal{H}$, then $\{ f_n(\cdot,\lambda) \}$ is a Cauchy sequence in $\mathcal{H}_\lambda$ for almost all $\lambda$.
the rest would be quite easy. But I don't know how to do it. Or maybe a completely different approach is needed?
Only a partial answer as this couldn't quite fit as a comment:
By Fubini-Tonelli's theorem: $$\|f\|_{\mathcal{H}}^2 = \iint_{\mathbb{C}^{n-1} \times (0,+\infty)} |f(z',\lambda)|^2 e^{-4\pi\lambda|z'|^2}\mathrm{d}m(z') \mathrm{d}\lambda = \|f\|^2_{L^2_\omega(\mathbb{C}^{n-1} \times (0,+\infty))}$$ where I denote by $\omega$ the function $(z',\lambda) \mapsto e^{-4\pi\lambda|z'|^2}$.
As a weighted $L^2$ space with positive measurable weight $\omega$, $L^2_\omega$ is complete, because it can be described as just $L^2(\mu)$ for a measure $\mu$ such that $\mathrm{d}\mu(z',\lambda) := \omega(z',\lambda)\mathrm{d}m(z')\mathrm{d}\lambda$.
As such, if we take a Cauchy sequence $(f_k)_k$ in $\mathcal{H}$, $(f_k)_k$ being a Cauchy sequence in $L^2_\omega$, there exists $f \in L^2_\omega$ such that $f_k \to f$ in $L^2_\omega$.
There does remain the question of the holomorphy of the $f(\cdot, \lambda)$s to show that $f$ is in $\mathcal{H}$ though.