Consider a family of compact metric spaces $(\Omega_t)_{t\in\mathbb{R}}$, and the corresponding product space $\Omega=\prod_{t\in\mathbb{R}}\Omega_t$. I will write its generic element as $\omega=(\omega_t)_{t\in\mathbb{R}}$. Among all functions $f:\Omega\rightarrow\mathbb{C}$, we may consider those that only depend on finitely many values of the parameter $t$, that is, \begin{equation} f(\omega)=\tilde{f}(\omega_{t_1},...,\omega_{t_n}). \end{equation} By Stone-Weierstrass, the completion of the space of such functions that are continuous, endowed with the usual uniform norm \begin{equation} \|f\|_\infty:=\max_{\omega\in\Omega}|f(\omega)|=\max_{\omega_{t_1},...,\omega_{t_n}}\left|\tilde{f}(\omega_{t_1},...,\omega_{t_n})\right|, \end{equation} should correspond exactly to the space of all continuous functions on $\Omega$ with the same norm.
I am wondering what happens if, instead, we consider some other norm. For instance, suppose that on $\Omega_t$ a measure $\mu_t$, and with it a natural notion of integration, is defined; as such, each finite product space $\Omega_{t_1}\times...\times\Omega_{t_n}$ carries with itself the corresponding product measure $\mu_{t_1,\dots,t_n}$. We could now endow the functions depending on finitely many coordinates (or a subspace of such functions) with the norm \begin{equation} \|f\|_p:=\left(\int\left|\tilde{f}(\omega_{t_1},\dots,\omega_{t_n})\right|^p\;\mathrm{d}\mu_{t_1,\dots,t_n}\right)^{1/p}. \end{equation} Could we somehow characterize the closure of this space in a similarly "satisfactory" way as the case of uniform norm (which, formally, would correspond to $p=\infty$)?
I apologize in advance if some of the assumptions above are not precise; examples with "sufficiently simple" choices of $\Omega$ (e.g., $\Omega_t=[0,1]$ and $\mu_t$ Lebesgue for all $t$) would be already much welcome.
Assuming each $\mu_t$ is a Radon probability (your seminorm $\|\cdot\|_p$ is ill-defined without some assumption on the total masses of the measures $\mu_t$), one can prove the existence and uniqueness of a Radon probability $\mu$ on $\Omega$ such that
$$\|f\|_p=\left(\int_\Omega|f(\omega)|^p\,d\mu(\omega)\right)^{1/p}$$
for every continuous function $f:\Omega\to\mathbb C$ which depends on only finitely many variables. Hence, the Hausdorff completion for the seminorm $\|\cdot\|_p$ of the space of continuous functions from $\Omega$ to $\mathbb C$ which depend on only finitely many variables may simply be described as the Banach space $L^p(\mu)$.
The existence and uniqueness of such a $\mu$ is essentially a quite ancient result of Daniell. While his original paper only deals with sequences of intervals of the real line, his approach can be easily adapted to arbitrarily large families of general compact Hausdorff spaces. You can find a treatment of this more general setting in Bourbaki's Integration, chap. III, §4, n°6, which also happens to deal with the case of general total masses other than just $1$. You can also check out n°5 for a similar result with complex Radon measures.