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Suppose $X:\Omega\to\mathbb{R}$ is an integrable random variable and the functions $f:\mathbb{R}\to\mathbb{R}$ and $f_i:\mathbb{R}\to\mathbb{R}$ are such that $f(X)$ and $f_i(X)$ is integrable for all $i\in\mathbb{N}$. Assume that $f_i$ converges to $f$ pointwise as $i\to\infty$.

What conditions do I need to impose on $\sup_{i\in\mathbb{N}} f_i$ to ensure that $\sup_{i\in\mathbb{N}} f_i(X)$ is also integrable? Would imposing $|\sup_{i\in\mathbb{N}} f_i(x)| < \infty$ for all $x\in\mathbb{R}$ ensure this?

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  • $\begingroup$ The property $\sup_{i\in \mathbb{N}} |f_i(x)|<\infty$ follows from the pointwise convergence, so can't imply anything. Regarding the question, even the uniform integrability is not sufficient. Say, $X$ is Cauchy, $f_i(x) = x\mathbf{1}_{[i-1,i]}(x)$. Then $(f_i(X),i\ge 1)$ is UI and $f_i(x) \to 0, i\to\infty$, but $\sup_i f_i(X) = X\mathbf{1}_{X\ge 0}$ is not integrable. WHy do you need the integrability of supremum in the first place? $\endgroup$
    – zhoraster
    Commented Jul 17, 2018 at 6:56

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