Prove that the following set is a dynkin system over R D = { B ∈ B ( R ) | ∀ ε &gt; 0 ∃ A ∈ F 2 μ ( A △ B ) &lt; ε }Where F 1 = { ( a , b ] ⊆ R | a ≤ b } F 2 = { ⋃ k = 1 m I k | I 1 , I 2 , . . . , I m ∈ F 1 }and μ is a probability measure of in measurable space ( R , B ( R ) )I am trying to prove that R ∈ D , but I am confused as it is clear that R ∉ F 2 and it cannot be applied that μ ( R △ R ) = 0 &lt; ε. Can you help me test that part? I have already proven that F 1 is a pi system, maybe that can help.

Question

Prove that the following set is a dynkin system over       R        D    =  {  B  ∈      B    (  R  )      |    ∀  ε  &amp;gt;  0    ∃  A  ∈            F        2      μ  (  A  △  B  )  &amp;lt;  ε  }Where            F        1    =  {  (  a  ,  b  ]  ⊆      R        |    a  ≤  b  }            F        2    =  {      ⋃          k      =      1              m            I    k        |        I    1    ,      I    2    ,  .  .  .  ,      I    m    ∈            F        1    }and   μ is a probability measure of in measurable space   (      R    ,      B    (      R    )  )I am trying to prove that       R    ∈      D  , but I am confused as it is clear that       R    ∉            F        2   and it cannot be applied that   μ  (      R    △      R    )  =  0  &amp;lt;  ε. Can you help me test that part? I have already proven that             F        1   is a pi system, maybe that can help.

Miriam Payne · Accepted Answer

μ  (      R    ∖  (  −  n  ,  n  ]  )  &amp;lt;  ϵ  if   n is large enough because       R    ∖  (  −  n  ,  n  ] decreases to empty set.

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

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