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

Serena Carpenter 2022-05-26 Answered
Prove that the following set is a dynkin system over R
D = { B B ( R ) | ε > 0 A F 2 μ ( A B ) < 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 < ε. Can you help me test that part? I have already proven that F 1 is a pi system, maybe that can help.
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Answers (1)

Miriam Payne
Answered 2022-05-27 Author has 10 answers
μ ( R ( n , n ] ) < ϵ if n is large enough because R ( n , n ] decreases to empty set.
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