$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? $X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $?

My feeling is that this is not necessarily true. But cannot come up with an example.

Can someone provide a counterexample or give a proof for this statement?
If $X_n$ are defined on the same probability space then the theorem is true and a proof is demonstrated in this question.

If $X_n$ are defined on different probability spaces then clearly $+$ must be defined on them for the question to have any meaning.

If $X$ is defined as the number of Farmer Brown's sheep in the paddock at 10am and $Y$ as the position of an electron in the CERN supercollider at the same time (for some frame of reference) then clearly it is not meaningful to speak of adding them.
If $X_n$ are defined on the same probability space then the theorem is true and a proof is demonstrated in this question.

If $X_n$ are defined on different probability spaces then clearly $+$ must be defined on them for the question to have any meaning.

If $X$ is defined as the number of Farmer Brown's sheep in the paddock at 10am and $Y$ as the position of an electron in the CERN supercollider at the same time (for some frame of reference) then clearly it is not meaningful to speak of adding them.