One may find a counting measure or trivial probability measure on $L^p$. However, is there any meaningful measure constructed on $L^p$, so that one may try to find a size of subsets of $L^p$? The same question goes for the $W^{k,p}(\Omega)$ spaces as well.

My attempt: $L^p$ (constructed on any measure space) is a topological vector space so we can construct Borel $\sigma$-algebra. We can think of next measure for $r$-balls: $$\mu_0(B_r(f))=r \quad(\text{or }r^2,r^3,e^r,\text{etc})$$ where $$B_r(f)=\{g\in L^p:\|f-g\|_{L^p}<r\}.$$ Then $\mu_0$ can be extended to a pre-measure on the ring $$\mathcal{B}:=\bigg\{\bigcup_{i\in I}B_{r_i}(f_i):r_i\geq 0,f_i\in L^p,\text{$B_{r_i}(f_i)$ are pairwise disjoint},\,\text{$I$ any index set}\bigg\}.$$ So we can use Carathéodory's extension theorem to find a measure $\mu$ on Borel $\sigma$-algebra on $L^p$. Is there any study on such $\mu$? Here I just needed that $L^p$ is a topological vector space. So it seems quite a general result, thus I suspect there is already a name for it, but I could not find any.

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