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Let $G$ be a group acting ercodically on a probability measure space $(X, \mu)$. Let $\Gamma$ be a countable dense subgroup of $G$. Is the action of $\Gamma$ also ergodic?

The case I am interested in is actually when $G$ is a Lie group acting smoothly on a compact submanifold of $\mathbb{R}^n$ with the normalized Lebesgue measure.

Here by ergodic action I mean that if $\mu(gA \Delta A) = 0$ for all $g \in G$, then $\mu(A) = 0$ or 1.

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No in general, yes for the case your are interested in.

For the "no" part: In the general case, this is lacking a assumption giving a "link" between the topology of $G$ and the action; here is an example of such a discontinuous action:

Let $G=\mathbb{Z}\oplus\mathbb{Z}\sqrt{2}\oplus \mathbb{Z}\sqrt{3}$, $\Gamma=\mathbb{Z}\oplus \mathbb{Z}\sqrt{2}$. We endow $G$ with the induced topology from $\mathbb{R}$. In this case, $\Gamma$ is a dense subgroup. Now consider an ergodic action of $G/\Gamma\simeq \mathbb{Z}\sqrt{3}$ on a measure space (for example, an irrational rotation on the circle). This gives an action of $G$ itself, which is ergodic, but the action of $\Gamma$ is trivial.

For the "yes" part: the action of $G$ on $X$ induces a unitary representation on $L^2(X,\mu)$ (the Koopman representation) $$\pi:G\to \mathcal{U}(L^2(X)),$$ $$\pi(f)(x)=f(g^{-1}x).$$ A useful property that may (or may not) have this representation is to be strongly continuous, i.e. $$g_n\to g\implies \forall f \in L^2(X), \;\| \pi(g_n)f-\pi(g)f\|\to 0.$$ The relevance of this property to ergodicity of dense subgroup is that in this case, the stabilizer of a characteristic function $f=1_A$ $$Stab(f)=\{ g \in G \, : \, \pi(g)f=f \},$$ is then a closed subgroup; so if a set $A$ is $\Gamma$-invariant, $f=1_A$ is $\Gamma$-invariant, so is also $\bar{\Gamma}=G$-invariant, so $\mu(A)\in \{0,1\}$.

In the case you are interested in, this property of strong continuity is satisfied; see for example Lemma 1.6 in those notes http://u.math.biu.ac.il/~solomyb/TEACH/17/GrAct/Lec7.pdf for a statement giving sufficient conditions and a sketch of proof.

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