If $X_1,X_2,...$ be independent random variables and $X_n\geq 0$ (not i.i.d.) and if $\sum_{n}E(X_n)<\infty$, Can we use Fubini's theorem and write $\sum_{n}E(X_n)=E(\sum_{n}X_n)$? (I'm confused because when writing expectations as integrals since random variables do not have same distribution, the $dp_i$ of integrals are different.)
And, why monotone convergence theorem implies $\sum_{n}E(X_n)=E(\sum_{n}X_n)$
This is true even without the assumption that $\sum EX_n <\infty$. Let $Y_k=\sum_{n=1}^{k} X_n$. Then $Y_k$'s are nonnegative measurable functions increasing to $Y=\sum_{n=1}^{\infty} X_n$. Monotone Convergence Theorem and linearity of expectation gives the result.