Prove that the following set is a dynkin system over $\mathbb{R}$

$\mathcal{D}=\{B\in \mathcal{B}(R)|\mathrm{\forall}\epsilon >0\phantom{\rule{1em}{0ex}}\mathrm{\exists}A\in {\mathcal{F}}_{2}\phantom{\rule{1em}{0ex}}\mu (A\mathrm{\u25b3}B)<\epsilon \}$

Where

${\mathcal{F}}_{1}=\{(a,b]\subseteq \mathbb{R}|a\le b\}$

${\mathcal{F}}_{2}=\{\bigcup _{k=1}^{m}{I}_{k}|{I}_{1},{I}_{2},...,{I}_{m}\in {\mathcal{F}}_{1}\}$

and $\mu $ is a probability measure of in measurable space $(\mathbb{R},\mathcal{B}(\mathbb{R}))$

I am trying to prove that $\mathbb{R}\in \mathcal{D}$, but I am confused as it is clear that $\mathbb{R}\notin {\mathcal{F}}_{2}$ and it cannot be applied that $\mu (\mathbb{R}\mathrm{\u25b3}\mathbb{R})=0<\epsilon $. Can you help me test that part? I have already proven that ${\mathcal{F}}_{1}$ is a pi system, maybe that can help.

$\mathcal{D}=\{B\in \mathcal{B}(R)|\mathrm{\forall}\epsilon >0\phantom{\rule{1em}{0ex}}\mathrm{\exists}A\in {\mathcal{F}}_{2}\phantom{\rule{1em}{0ex}}\mu (A\mathrm{\u25b3}B)<\epsilon \}$

Where

${\mathcal{F}}_{1}=\{(a,b]\subseteq \mathbb{R}|a\le b\}$

${\mathcal{F}}_{2}=\{\bigcup _{k=1}^{m}{I}_{k}|{I}_{1},{I}_{2},...,{I}_{m}\in {\mathcal{F}}_{1}\}$

and $\mu $ is a probability measure of in measurable space $(\mathbb{R},\mathcal{B}(\mathbb{R}))$

I am trying to prove that $\mathbb{R}\in \mathcal{D}$, but I am confused as it is clear that $\mathbb{R}\notin {\mathcal{F}}_{2}$ and it cannot be applied that $\mu (\mathbb{R}\mathrm{\u25b3}\mathbb{R})=0<\epsilon $. Can you help me test that part? I have already proven that ${\mathcal{F}}_{1}$ is a pi system, maybe that can help.