Prove that the following set is a dynkin system over $\mathbb{R}$ $$\mathscr{D} = \{B\in \mathscr{B}(R)| \forall \varepsilon >0 \quad \exists A\in \mathscr{F}_2 \quad \mu (A\triangle B) < \varepsilon \}$$ Where $$\mathscr{F}_1 = \{(a,b]\subseteq \mathbb{R}|a\leq b \}$$ $$\mathscr{F}_2 = \{\bigcup_{k=1}^{m}I_k | I_1, I_2,..., I_m\in \mathscr{F}_1 \} $$ and $ \mu $ is a probability measure of in measurable space $ (\mathbb {R}, \mathscr {B} (\mathbb {R})) $
I am trying to prove that $ \mathbb {R} \in \mathscr {D} $, but I am confused as it is clear that $ \mathbb {R} \notin \mathscr {F}_2 $ and it cannot be applied that $ \mu (\mathbb {R} \triangle \mathbb {R}) = 0 <\varepsilon $. Can you help me test that part? I have already proven that $\mathscr{F}_1$ is a pi system, maybe that can help.
$\mu (\mathbb R \setminus (-n,n]) <\epsilon$ if $n$ is large enough because $\mathbb R \setminus (-n,n]$ decreases to empty set.