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If $X_n$ are independent random variables, then does $\sum_n \mathbb{E}(X_n)=\mathbb{E}(\sum_n X_n)$?

This is not a homework problem but rather a question I had. If it is not true, what are the weakest conditions that would make it true? I am aware that if either are absolutely summable, then fubini tells us the answer is yes. I am more interested in the case that fubini does not hold. What weaker conditions can we impose but use the independence of the random variables?

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    $\begingroup$ Indeed, your proposition holds regardless of the dependence of the random variables. For a more general and complete argument you may wanna see the question tagged above. $\endgroup$
    – Nighty
    Commented Oct 21, 2014 at 7:29
  • $\begingroup$ I am more interested when the conditions of fubini aren't satisfied. Can the independence give us new weaker conditions? $\endgroup$
    – anonymous
    Commented Oct 21, 2014 at 8:46
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    $\begingroup$ @DavideGiraudo Why did you see fit to reopen this obvious duplicate? $\endgroup$
    – Did
    Commented Oct 21, 2014 at 9:51
  • $\begingroup$ @Did I should have red more carefully the revision: it is still a duplicate after this revision. $\endgroup$
    – Davide Giraudo
    Commented Oct 21, 2014 at 14:35

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