model theory – Roelcke precompactness and Ramsey property

A survey by Nguyen Van Thé (2014) has Conjecture 1,
which is that
“every closed oligomorphic
subgroup of $S_∞$ should have a metrizable universal minimal flow with a generic
orbit.” Later, it goes on to say that “it is even possible that this should be
true for a larger class of groups, called Roelcke precompact.” (Let me call this Conjecture 1′.) Now, Kwiatkowska (2018) exhibited a group without a metrizable universal minimal flow that is not Roelcke precompact, so we need to stick to Conjecture 1.

How about the converse of Conjecture 1′, i.e., the statement that a closed subgroup of $S_∞$ is Roelcke precompact if it has a metrizable universal minimal flow? Is there a proof or a counterexample? In absence of either of the two, do people believe it?

real analysis – Generalization of Kolmogorov precompactness criterion

In the book Elements in functional analysis from Hirsch and Lacombe, the Kolmogorov precompactness criterion for families of $L^p$ functions is stated as follows:

Theorem: Let $H subseteq L^p(mathbb{R}^d)$, with $p in (1,infty)$. Then $H$ is precompact if and only if the following conditions hold:

  1. $H$ is bounded w.r.t. the norm $||cdot||_{L^p(mathbb{R}^d)}$;
  2. $lim_{R to +infty} sup_{f in H} int_{B_R^c} |f(x)|^p dx = 0$, where $B_R^c$ denotes the complement in $mathbb{R}^d$ of the ball of radius $R$;
  3. $lim_{|a| to 0} sup_{f in H} ||tau_af-f||_{L^p(mathbb{R}^d)}=0$, where $tau_a$ denotes the translation operator.

Now, I think that one can modify a little bit this statement in order to make it valid also when we consider families of functions in $L^p(Omega)$, where $Omega$ is an open set in $mathbb{R}^d$:

Claim: Let $Omega$ be an open set in $mathbb{R}^d$, let $p in (1,infty)$, and let $H subseteq L^p(Omega)$. For all $R > 0$, define
$$Omega_R = {x in Omega: text{dist}(x, {Omega}^c) > frac{1}{R}} cap B(0,R).$$
Then $H$ is precompact if and only id the following holds

  1. $H$ is bounded w.r.t. the norm $||cdot||_{L^p(Omega)}$;
  2. $lim_{R to +infty} sup_{f in H} int_{Omega setminus Omega_R} |f(x)|^p dx = 0$;
  3. $lim_{|a| to 0} sup_{f in H} ||tau_af-f||_{L^p(Omega_R)}=0$ for all $R >0$.

I’m trying to prove this last equivalence using the theorem I’ve written above. So far this is my strategy: First, notice that we can look at $L^p(Omega)$ as a subset of $L^p(mathbb{R}^d)$ extending every function $f$ in $L^p(Omega)$ to a function $tilde{f}$ in $L^p(mathbb{R}^d)$ in this way:
$$tilde{f}(x) =begin{cases}
f(x) & x in Omega \
0 & text{otherwise}
end{cases}$$

Then $H$ is precompact in $L^p(Omega)$ if and only if it’s precompact in $L^p(mathbb{R}^d)$.

If I prove that conditions 1,2,3 of the claim are equivalent to conditions 1,2,3 of the theorem, I’m done. Condition 1 is easy, but I can’t really see how conditions 2 and 3 of the theorem should be equivalent to condition 2 and 3 of the claim.

Any help, remark or suggestion is appreciated, thank you.