Generalization of Kolmogorov precompactness criterion

Question

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 conditions 2 and 3 of the claim.

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