Let $Z$ be a lower semi integrable r.v. and $W_n$ a sequence of bounded r.v increasing and dominated by another bounded r.v from above $W_n \leq W$, why is true that $E[ZW_n]$ converges to $E[ZW]$?
The negative part $Z^-$ is taken care of by the standard Lebesgue convergence theorem, but the positive part $Z^+W_n$ I don't know how to make it converge since it may be negative.
This was used in the book of Paolo Baldi: Stochastic Calculus page 88. He uses the monotone class theorem to prove that to find the conditional expectation it is sufficient to check for a class generating the sigma algebra, stable with respect to finite intersections.
Thanks.
Taking the lebesgue measure in $[0,1]$, $\lambda$, $Z=\dfrac{1}{x}$, $W_n = -\dfrac{1}{n}$, one can see that the integral sequence doesn't even exist.
It seems that Remark 4.2 refers to integrable random variables $X$ and $Z$. Proceeding with your example, where $([0,1],\mathcal{B}_[0,1],\lambda)$ is the prob. space and $Z(\omega)=\omega^{-1}\times 1\{\omega>0\}$, take $W_n(\omega)=-1_{[0,n^{-1})}(\omega)$. Then $Z$ is l.s.i., $-1\le W_n\le 0$, and $W_n\nearrow 0$. However, $$ 0=\mathsf{E}WZ\ne\lim_{n\to\infty}\mathsf{E}W_nZ=-\infty. $$