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As I asserted in my comments, I think it is too much to hope for a reasonable topology on the space of random variables which makes the map $X \mapsto E(X)$ continuous. This would be a bit like asking for a topology on $L^1(\mathbb{R})$ which makes $f \mapsto \int f$ continuous: there are just too many weird examples where you can't swap the order of limit and integration.

But all is not lost. In analysis one salvages the situation by making stronger assumptions: for instance, if $f_n \to f$ pointwise and $|f_n(x)| \leq g(x)$ for some integrable function $g$ and all $n$, then $\int f_n \to \int f$ (the dominated convergence theorem). There is a sort of counterpart to this in probability theory.

Definition: A sequence $X_n$ of random variables is uniformly integrable if:

• $E(|X_n|)$ is uniformly bounded in $n$: there is a constant $K$ such that $E(|X_n|) \leq K$ for all $n$
• For every $\varepsilon > 0$ there exists $\delta > 0$ such that $\int_A |X_n| < \varepsilon$ for all $n$ whenever $P(A) < \delta$

Uniform integrability is implied by the stronger (but more easily checked) condition that $E(|X_n|^{1 + \delta})$ is uniformly bounded for some $\delta > 0$.

Theorem: Suppose $X_n$ converges to $X$ in distribution and the sequence $|X_n|^k$ is uniformly integrable. Then $E(X_n^j) \to E(X^j)$ whenever $1 \leq j \leq k$. (Reference)

There are some other results of this flavor in the wikipedia page on convergence of random variables but this theorem is the best result that I know.

As I asserted in my comments, I think it is too much to hope for a reasonable topology on the space of random variables which makes the map $X \mapsto E(X)$ continuous. This is a bit like hoping that pointwise convergence or convergence in measure implies convergence in the $L^1$ norm; it seems reasonable, but there are simple counter-examples.

But all is not lost. In analysis one salvages the situation by making stronger assumptions: for instance, if $f_n \to f$ pointwise and $|f_n(x)| \leq g(x)$ for some integrable function $g$ and all $n$, then $\int f_n \to \int f$ (the dominated convergence theorem). There is a sort of counterpart to this in probability theory.

Definition: A sequence $X_n$ of random variables is uniformly integrable if:

• $E(|X_n|)$ is uniformly bounded in $n$: there is a constant $K$ such that $E(|X_n|) \leq K$ for all $n$
• For every $\varepsilon > 0$ there exists $\delta > 0$ such that $\int_A |X_n| < \varepsilon$ for all $n$ whenever $P(A) < \delta$

Uniform integrability is implied by the stronger (but more easily checked) condition that $E(|X_n|^{1 + \delta})$ is uniformly bounded for some $\delta > 0$.

Theorem: Suppose $X_n$ converges to $X$ in distribution and the sequence $|X_n|^k$ is uniformly integrable. Then $E(X_n^j) \to E(X^j)$ whenever $1 \leq j \leq k$. (Reference)

There are some other results of this flavor in the wikipedia page on convergence of random variables but this theorem is the best result that I know.

As I asserted in my comments, I think it is too much to hope for a reasonable topology on the space of random variables which makes the map $X \mapsto E(X)$ continuous. This would be a bit like asking for a topology on $L^1(\mathbb{R})$ which makes $f \mapsto \int f$ continuous: there are just too many weird examples where you can't swap the order of limit and integration.

But all is not lost. In analysis one salvages the situation by making stronger assumptions: for instance, if $f_n \to f$ pointwise and $|f_n(x)| \leq g(x)$ for some integrable function $g$ and all $n$, then $\int f_n \to \int f$ (the dominated convergence theorem). There is a sort of counterpart to this in probability theory.

Definition: A sequence $X_n$ of random variables is uniformly integrable if:

• $E(|X_n|)$ is uniformly bounded in $n$: there is a constant $K$ such that $E(|X_n|) \leq K$ for all $n$
• For every $\varepsilon > 0$ there exists $\delta > 0$ such that $\int_A |X_n| < \varepsilon$ for all $n$ whenever $P(A) < \delta$

Uniform integrability is implied by the weaker (and more easily checked) condition that $E(|X_n|^{1 + \delta})$ is uniformly bounded for some $\delta > 0$.

Theorem: Suppose $X_n$ converges to $X$ in distribution and the sequence $|X_n|^k$ is uniformly integrable. Then $E(X_n^j) \to E(X^j)$ whenever $1 \leq j \leq k$. (Reference)

There are some other results of this flavor in the wikipedia page on convergence of random variables but this theorem is the best result that I know.

As I asserted in my comments, I think it is too much to hope for a reasonable topology on the space of random variables which makes the map $X \mapsto E(X)$ continuous. This would be a bit like asking for a topology on $L^1(\mathbb{R})$ which makes $f \mapsto \int f$ continuous: there are just too many weird examples where you can't swap the order of limit and integration.

But all is not lost. In analysis one salvages the situation by making stronger assumptions: for instance, if $f_n \to f$ pointwise and $|f_n(x)| \leq g(x)$ for some integrable function $g$ and all $n$, then $\int f_n \to \int f$ (the dominated convergence theorem). There is a sort of counterpart to this in probability theory.

Definition: A sequence $X_n$ of random variables is uniformly integrable if:

• $E(|X_n|)$ is uniformly bounded in $n$: there is a constant $K$ such that $E(|X_n|) \leq K$ for all $n$
• For every $\varepsilon > 0$ there exists $\delta > 0$ such that $\int_A |X_n| < \varepsilon$ for all $n$ whenever $P(A) < \delta$

Uniform integrability is implied by the stronger (but more easily checked) condition that $E(|X_n|^{1 + \delta})$ is uniformly bounded for some $\delta > 0$.

Theorem: Suppose $X_n$ converges to $X$ in distribution and the sequence $|X_n|^k$ is uniformly integrable. Then $E(X_n^j) \to E(X^j)$ whenever $1 \leq j \leq k$. (Reference)

There are some other results of this flavor in the wikipedia page on convergence of random variables but this theorem is the best result that I know.

As I asserted in my comments, I think it is too much to hope for a reasonable topology on the space of random variables which makes the map $X \mapsto E(X)$ continuous. This would be a bit like asking for a topology on $L^1(\mathbb{R})$ which makes $f \mapsto \int f$ continuous: there are just too many weird examples where you can't swap the order of limit and integration.

But all is not lost. In analysis one salvages the situation by making stronger assumptions: for instance, if $f_n \to f$ pointwise and $|f_n(x)| \leq g(x)$ for some integrable function $g$ and all $n$, then $\int f_n \to \int f$ (the dominated convergence theorem). There is a sort of counterpart to this in probability theory.

Definition: A sequence $X_n$ of random variables is uniformly integrable if:

• E(|X_n|) is uniformly bounded in $n$: there is a constant $K$ such that $E(|X_n|) \leq K$ for all $n$
• For every $\varepsilon > 0$ there exists $\delta > 0$ such that $\int_A X_n < \varepsilon$ for all $n$ whenever $P(A) < \delta$

Uniform integrability is implied by the weaker (and more easily checked) condition that $E(|X_n|^{1 + \delta})$ is uniformly bounded for some $\delta > 0$.

Theorem: Suppose $X_n$ converges to $X$ in distribution and the sequence $|X_n|^k$ is uniformly integrable. Then $E(X_n^j) \to E(X^j)$ whenever $1 \leq j \leq k$. (Reference)

There are some other results of this flavor in the wikipedia page on convergence of random variables but this theorem is the best result that I know.

As I asserted in my comments, I think it is too much to hope for a reasonable topology on the space of random variables which makes the map $X \mapsto E(X)$ continuous. This would be a bit like asking for a topology on $L^1(\mathbb{R})$ which makes $f \mapsto \int f$ continuous: there are just too many weird examples where you can't swap the order of limit and integration.

But all is not lost. In analysis one salvages the situation by making stronger assumptions: for instance, if $f_n \to f$ pointwise and $|f_n(x)| \leq g(x)$ for some integrable function $g$ and all $n$, then $\int f_n \to \int f$ (the dominated convergence theorem). There is a sort of counterpart to this in probability theory.

Definition: A sequence $X_n$ of random variables is uniformly integrable if:

• $E(|X_n|)$ is uniformly bounded in $n$: there is a constant $K$ such that $E(|X_n|) \leq K$ for all $n$
• For every $\varepsilon > 0$ there exists $\delta > 0$ such that $\int_A |X_n| < \varepsilon$ for all $n$ whenever $P(A) < \delta$

Uniform integrability is implied by the weaker (and more easily checked) condition that $E(|X_n|^{1 + \delta})$ is uniformly bounded for some $\delta > 0$.

Theorem: Suppose $X_n$ converges to $X$ in distribution and the sequence $|X_n|^k$ is uniformly integrable. Then $E(X_n^j) \to E(X^j)$ whenever $1 \leq j \leq k$. (Reference)

There are some other results of this flavor in the wikipedia page on convergence of random variables but this theorem is the best result that I know.