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Suppose we have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, some $\sigma$-algebra $\mathcal{G} \subseteq \mathcal{F}$ and a random variable $X$ which is $\mathcal{F}$-measurable. Suppose that $\mathbb{E}(|X|^2) < \infty$. Can we bound $|\mathbb{E}(X | \mathcal{G})|$ almost everywhere in terms of $X$? If not, do we need extra assumptions on $X$, for example $\mathbb{E}(|X|^n) < \infty$ for a large enough $n \in \mathbb{N}$?

For context: I want to use the Dominated Convergence theorem on $\mathbb{E}(X|\mathcal{G})$, but for that I need a dominating function. If there is one, I suspect it will be a function depending on $X$.

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