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Math Help - Measure of countable co-countable set

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    Measure of countable co-countable set

    Let X be an uncountable set, let  E \subset X
    Consider the  \sigma -algebra  \mathbb {M} = \{ E \ countable \ or \ E^c \ countable \}

    Define  \mu : \mathbb {M} \rightarrow [0, \infty ] by  \mu (E) = 0 \ if \ E \ countable and  \mu (E) = 0 \ if \ E^c \ countable

    Prove that  \mu is a measure.

    Proof so far.

    Let E_1,E_2, ... \in \mathbb {M} , claim:  \mu ( \bigcup _{n=1}^ \infty \mu (E_n) = \sum ^ \infty _{n=1} \mu (E_n)

    Case 1) If E_n is countable for all n, then  \bigcup _{n=1}^ \infty \mu (E_n) is countable and  \mu ( \bigcup _{n=1}^ \infty \mu (E_n) = 0 = \sum ^ \infty _{n=1} \mu (E_n) since  \mu (E_n) = 0 \ \ \ \forall n

    Case 2) If E_n ^c is countable. Since  ( \bigcup _{n=1}^ \infty  (E_n) )^c \subset \bigcup ^ \infty _{n=1} E_n^c and the latter is countable, I then have  \mu ( \bigcup _{n=1}^ \infty  (E_n) ) = 1

    But then I also have  \sum ^ \infty _{n=1} \mu (E_n) = \mu (E_1) + \mu (E_2) = . . . = 1 + 1 + 1 + . . . = \infty

    So the two ain't equal... What did I do wrong?

    Case 3) Either E_n is countable or E_n^c is countable. How should I proceed with this one?

    Thank you!
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let X be an uncountable set
    (....)
    Let E_1,E_2, ... \in \mathbb {M} be disjoint subsets, claim:  \mu ( \bigcup _{n=1}^ \infty E_n) = \sum ^ \infty _{n=1} \mu (E_n)

    Case 2) If E_n ^c is countable. Since  ( \bigcup _{n=1}^ \infty  (E_n) )^c \subset \bigcup ^ \infty _{n=1} E_n^c and the latter is countable, I then have  \mu ( \bigcup _{n=1}^ \infty  (E_n) ) = 1

    But then I also have  \sum ^ \infty _{n=1} \mu (E_n) = \mu (E_1) + \mu (E_2) = . . . = 1 + 1 + 1 + . . . = \infty

    So the two ain't equal... What did I do wrong?

    Case 3) Either E_n is countable or E_n^c is countable. How should I proceed with this one?
    You can't expect equality if the subsets aren't disjoint.

    The key is given by the following question: is it possible that two sets E_1,E_2 are disjoint if they are such that E_1^c and E_2^c are countable?
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