I know what you mean, but I cannot show it that way. I have been told that linearity only applies to finite sums.
I may need to use a convergence theorem (a random var. convergence theorem, maybe), but still don't know how to show this.
I should have said all this in the first place, sorry .
You're right, a theorem is needed in your case. Namely Lebesgue's dominated convergence theorem (or whatever name you give to it). Let's check it applies here:
a) for every n, and the right-hand side is integrable because of your hypothesis;
b) almost surely, converges to (this is a consequence of the inequality above, the fact that absolute convergence of a series implies convergence, and the property that implies almost-surely).
As a consequence of these two facts, you can tell, with the dominated convergence theorem, that indeed . Notice at last that , so that you can also write the previous limit as . And this is what you wanted.