Consider a discrete finite set $\Omega=X\times Y \in \mathbb{R}^{m\times n}$ for finite $m,n$. Let $(\Omega,\Sigma,\mu)$ be the measure space. ($\Sigma$ is the power set and $\mu$ is $\sigma$-finite measure.)
Let $\mathcal{X}$ be the set of all measurable functions on the above measure space. Clearly $|\mathcal{X}|$ (cardinality or size of $\mathcal{X}$) is exponential in $|\Omega|$.
Question 1: Can we come up with a dense subset of $\mathcal{X}$ such that we can get better complexity in the dimensions? I mean do we have something of VC-dimension or Covering number sorts for $\mathcal{X}$ that we can rely on which is way better than directly using $|\mathcal{X}|$?
Or let's consider a subset of $\mathcal{X}$. That is, consider the lebesgue measurable space $L_p(\mu)\subseteq \mathcal{X}$ for $p\geq 1$.
Question 2: Is there a VC-dimension or Covering number sorts for $L_p(\mu)$ spaces in literature?
I spent last couple of hours (only to end up writing this post at 3am!) researching on this. All I could see were unsatisfactory. For example:
From paper titled "The Learning Rates of Regularized Regression Based on Reproducing Kernel Banach Spaces": (I don't know how big $s$ would be for $L_p(\mu)$ spaces!)
