Working through the book Brownian Motion & Stochastic Calculus by Revuz and Yor, I got confronted with the following problem:
Let $X$ be a random variable s.t. $ E[X^+] < \infty $.
Now assume that using the integrability of $\max(X,-n), \ \forall n \in \mathbb{N} $ we were able to show that $\max(X,-n) \leq E[\max(X,-n) \mid \mathcal{F} \ ], \ \forall n \in \mathbb{N} $.
Can we somehow deduce from this that also \ X \leq E[X | \mathcal{F} ] ?
Or in other words, does $E[\max(X,-n) \ | \ \mathcal{F} \ ] \ $ tend to E[X \mid \mathcal{F} ] \ as $ n \rightarrow + \infty $ ? (which would of course imply the above inequality)
Can we maybe somehow apply monotone convergence or dominated convergence for conditional expectations?
Thanks a lot for any help towards a solution!