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**tttcomrader** Let X be an uncountable set

(....)

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

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

But then I also have $\displaystyle \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 $\displaystyle E_n$ is countable or $\displaystyle E_n^c$ is countable. How should I proceed with this one?