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Here is the problem: enter image description here

Given $X, X_1, X_2, \ldots$, non-negative random variable with finite expectation and $X_n \to X$ pointwise and $Y_{n,K} = \min(X_n,K)$, we are asked to see if

a) $\lim_{K \to \infty} \lim_{n \to \infty} E(Y_{n,K}) = E(X)$

and

b) $\lim_{n \to \infty} \lim_{K \to \infty} E(Y_{n,K}) = E(X)$.

For the first problem, here is my attempt.

For fixed $K$, $\min(X_n, K) \leq K$. So by the dominated convergence theorem, $$\lim_{n \to \infty} E(Y_{n,K}) = E(\lim_{n \to \infty} \min(X_n,K)) = E(\min(X,K)).$$

And hence $$\lim_{K \to \infty} \lim_{n \to \infty} E(Y_{n,K}) = E(X).$$

The second part, for fixed $n$, $Y_{n,K} \leq X_n$, with $X_n$ having a finite expectation. So $$\lim_{K \to \infty} E(Y_{n,K}) = E(X_n),$$ and so $$\lim_{n \to \infty} \lim_{K \to \infty} E(Y_{n,K}) = E(X).$$

Is this correct? Seems like I missed something.

Source: A First Course in Rigorous Probability Theory by Jeffrey Rosenthal.

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    $\begingroup$ The random variables $X_n:(\mathbb R, \mathcal{B}(\mathbb{R}), \lambda)\to\mathbb R$ (Lebesgue measure) given by $X(x) = nI(0\lt x \le 1/n)$ are worth contemplating. Each has unit expectation but their pointwise limit is the zero function, with zero expectation. $\endgroup$
    – whuber
    Dec 7, 2022 at 4:38
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    $\begingroup$ @whuber was the part $\lim_{n \to \infty} \min(X_n, K) = \min(X,K)$ wrong? $\endgroup$
    – Phil
    Dec 7, 2022 at 4:48

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