# Prove that the following set is a dynkin system over <mrow class="MJX-TeXAtom-ORD"> <mi mat

Prove that the following set is a dynkin system over $\mathbb{R}$
$\mathcal{D}=\left\{B\in \mathcal{B}\left(R\right)|\mathrm{\forall }\epsilon >0\phantom{\rule{1em}{0ex}}\mathrm{\exists }A\in {\mathcal{F}}_{2}\phantom{\rule{1em}{0ex}}\mu \left(A\mathrm{△}B\right)<\epsilon \right\}$
Where
${\mathcal{F}}_{1}=\left\{\left(a,b\right]\subseteq \mathbb{R}|a\le b\right\}$
${\mathcal{F}}_{2}=\left\{\bigcup _{k=1}^{m}{I}_{k}|{I}_{1},{I}_{2},...,{I}_{m}\in {\mathcal{F}}_{1}\right\}$
and $\mu$ is a probability measure of in measurable space $\left(\mathbb{R},\mathcal{B}\left(\mathbb{R}\right)\right)$

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 $\mu \left(\mathbb{R}\mathrm{△}\mathbb{R}\right)=0<\epsilon$. Can you help me test that part? I have already proven that ${\mathcal{F}}_{1}$ is a pi system, maybe that can help.
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Miriam Payne
$\mu \left(\mathbb{R}\setminus \left(-n,n\right]\right)<ϵ$ if $n$ is large enough because $\mathbb{R}\setminus \left(-n,n\right]$ decreases to empty set.