Two questions on the existence of sets associated with LaSalle invariance principle I am copying in verbatim a standard reference on Lasalle's theorem (which is Hassan Khalil's textbook, page 128)
Theorem:

Let $\Omega \subset D$ be a compact set that is positively invariant
with respect to an ODE $\dot x = f(x)$. Let $V: D \to \mathbb{R}$ be a
continuously differentiable function such that $\dot V(x) \leq 0$ in
$\Omega$. Let $E$ be the set of all points in $\Omega$ where $\dot
 V(x) = 0$. Let $M$ be the largest invariant set in $E$. Then every
solution trajectory starting in $\Omega$ approaches $M$ as $t \to
 \infty$.

Here are my two questions about this theorem:

How would you know if there exists such a positively invariant set $\Omega$ in the first place? And how would you construct this set $\Omega$?

How would you know there exists an invariant set $M$ in $E$ in the first place? And how would you construct this set $M$?


Thank you.

For added context, this question involves an extremely well studied/known theorem in dynamical system theory. The text in reference is cited 40,000 times, hence not some obscure result. However, very little is said about the existence (or construction) of certain sets used in this theorem and most examples do not verify the existence of such sets, or does so in a case-by-case basis with no universal method of constructing these sets.
LaSalle's principle assumes the existence of $\Omega$, so you don't need to construct it to prove the theorem. If you are trying to apply the theorem, then usually there's an obvious candidate for $\Omega$, such as an equilibrium point, bounded set containing the solution, etc.

The null set is (trivially) always an invariant set in $E$. To construct $M$, let $\mathcal{M}$ be the set of all invariant subsets of $E$ (nonempty because $\varnothing\in \mathcal{M}$) and set $M = \cup \mathcal{M}$. Again, it is usually easy to tell what $M$ is (e.g. an equilibrium) from the specific problem.
LaSalle's principle assumes the existence of $\Omega$, so you don't need to construct it to prove the theorem. If you are trying to apply the theorem, then usually there's an obvious candidate for $\Omega$, such as an equilibrium point, bounded set containing the solution, etc.

The null set is (trivially) always an invariant set in $E$. To construct $M$, let $\mathcal{M}$ be the set of all invariant subsets of $E$ (nonempty because $\varnothing\in \mathcal{M}$) and set $M = \cup \mathcal{M}$. Again, it is usually easy to tell what $M$ is (e.g. an equilibrium) from the specific problem.