Supremum of conditional expectations

Question

Let $X_{n \in \mathbb{N}}$ be a sequence of real valued random variables defined on a filtered probability space $(\Omega,F,P)$ with filtration $F_{n}\subseteq F$ and the $X_{n}$ being not adapted. If we have: $$\mathbb{E}(\underset{n}{\sup} |X_{n}|)<\infty$$ Does it hold true that: $$\mathbb{E}(\underset{n}{\sup} (\mathbb{E}(X_{n}|F_{n})))<\infty$$ ?

Answer 1

If I further assume $b:=\mathbb{E}((\underset{i \in \mathbb{N}}{\sup} X_{i})^{2})<\infty$ then i have by an application of Doobs Maximal Inequality: $$\mathbb{E}(\underset{k\leq n}{\max}\mathbb{E}(X_{k}|F_{k}))\leq \mathbb{E}(\underset{k\leq n}{\max}\mathbb{E}(\underset{i \in \mathbb{N}}{\sup} X_{i}|F_{k}))\leq \mathbb{E}(\underset{k\leq n}{\max}|\mathbb{E}(\underset{i \in \mathbb{N}}{\sup} X_{i}|F_{k})|)\leq 1 + \mathbb{E}((\underset{k\leq n}{\max}|\mathbb{E}(\underset{i \in \mathbb{N}}{\sup} X_{i}|F_{k})|)^{2})\overset{\text{DoobsMaximalIneq.}}{\leq}1 + 4 \mathbb{E}((\mathbb{E}(\sup X_{i}|F_{n}))^{2})\overset{\text{JensenIneqCond.}}{\leq}1 + 4 \mathbb{E}(\mathbb{E}((\sup X_{i})^{2}|F_{n}))= 1 + 4 \mathbb{E}((\sup X_{i})^{2})= 1 + 4b <\infty$$ So $\mathbb{E}(\underset{k\leq n}{\max}\mathbb{E}(X_{k}|F_{k}))$ is bounded and converges by monotone convergence to $\mathbb{E}(\underset{i \in \mathbb{N}}{\sup} \mathbb{E}(X_{i}|F_{i}))$ which is also bounded by 1+4b cause it holds true for every member of the sequence.