A question on ultraproducts of $L_{p}(\mu)$-spaces

Question

Let $I$ be a set, $\mathcal{U}$ be an ultrafilter on $I$ and $1\leq p<\infty$. Let $X_{i}=L_{p}(\mu_{i})(i\in I)$, where $\mu_{i}$ is a probability measure for each $i\in I$. Relying on standard results from Banach lattice theory, we can prove that the ultraproduct $(X_{i})_{\mathcal{U}}$ of $(X_{i})_{i\in I}$ is isometrically isomorphic to $L_{p}(\mu)$ for some measure $\mu$.

Question 1: Can the above measure $\mu$ be taken to be finite or even a probability measure?

Question 2: Is there a direct construction of the measure $\mu$ without relying on Banach lattice theory?

Thank you!

Answer 1

As for the first question the answer is no even if $\mu$ is assumed to be $\sigma$-finite as by a simple change of measure you may reduce the problem to the case where $\mu$ is finite. Let us fix a finite measure $\mu$.

When $p=1$ we may use the following reasoning. The space $L_1(\mu)$ is weakly compactly generated because the inclusion map ${\rm id}\colon L_2(\mu)\to L_1(\mu)$ has dense range. On the other hand, it is well-known that $\ell_1(\Gamma)$ does not embed into a weakly compactly generated space for any uncountable set $\Gamma$, however by the principle of local reflexivity you will find it in some ultrapower of $L_1[0,1]$.

When $p\in (1,2)$, then $\ell_p(\Gamma)$ does not embed into $L_p(\mu)$ with $\mu$ finite for $\Gamma$ uncountable; this is proved in:

P. Enflo, H. P. Rosenthal, Some results concerning $L_p(\mu)$-spaces. J. Functional Analysis 14 (1973), 325–348.

This was extended to $p\in (2,\infty)$ by Bill Johnson and Gideon Schechtman (see this paper).

As for the second question, should such construction exist, it is most likely covered in one of Fremlin's Measure theory volumes.