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Montenovofe 2022-07-02 Answered
Let T be a countable set, A = { A T : A  is finite or  A c  is finite } and μ : A [ 0 , ) definided for every A A by
μ ( A ) = { 0 ,  if  A  is finite  1 ,  if  A c  is finite. 
Then every set A A , such that A c is finite, is an atom with respect of μ.
But I can't see how it is concluded that every set A A , such that A c is finite, is an atom with respect of μ, can someone help me?
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Answers (1)

Maggie Bowman
Answered 2022-07-03 Author has 14 answers
If A c is finite, then A must be infinite. Otherwise, T = A A c would be finite. So μ ( A ) = 1. Now, if E A and E A , then, by the latter condition, E is finite and satisfies μ ( E ) = 0, or E c is finite, which , again. implies that E is infinite and μ ( E ). So either μ ( E ) = μ ( A ) or μ ( E ) = 0 for all E A with E A , which is exactly the definition of A being an atom.
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