Conditional expectation and limit

Question

Let $F$ be a $\sigma$-algebra.

I'm wondering if $E^{F}[\sum_{n=0}^{+\infty} X_{n}] = \sum_{n=0}^{+\infty} E^{F}[X_{n}]$ with for exemple $X_{n}$ non negative mesurable function.

Thank you so much for your help!

Answer 1

Surely true. You can verify it from definition of conditional expectations using Monotone Convergence Theorem.