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:*

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

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