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I'm interested in a new (to me) mode of convergence which is stronger than convergence in measure/probability. I want to know if it has a name and if it is used much in the literature. I will write my question twice using both probability theory notation and measure theory notation to ensure that I reach the correct audience.


Question in measure theory notation

Let $f_n:\mathbb{X} \to \mathbb{Y}$ be a sequence of measurable functions where $(\mathbb{X},d_\mathbb{X})$ and $(\mathbb{Y},d_\mathbb{Y})$ are complete separable metric spaces (if it helps, assume $\mathbb{X}=\mathbb{Y}=\mathbb{R}$). Let $\mu$ be a probability measure on $\mathbb{X}$.

It is well-known that $f_n$ converges in measure to a measurable function $f_n:\mathbb{X} \to \mathbb{Y}$ if $\mu \{x:d_\mathbb{Y}(f_n, f) > \varepsilon\} \to 0$ as $n \to \infty$ for all $\varepsilon > 0$.

Recall, that every function $f$ induces a $\sigma$-algebra $$\sigma(f) = \{A \subseteq \mathbb{X} | A = f^{-1}(B), B\subseteq \mathbb{Y} \text{ measurable}\}.$$ My new mode of convergence will require that $f_n \to f$ in measure and that (informally) $\sigma(f_n) \to \sigma(f)$. This latter condition can be made formal using conditional expectation as follows.

Recall that the conditional expectation of a measurable function $g:\mathbb{X} \to \mathbb{R}$ conditioned on the sigma-algebra $\sigma(f)$, written $E(g|\sigma(f))$ or just $E(g|f)$ is the ($\mu$-a.e.) unique $\sigma(f)$-measurable function $h$ given by $$\int_A h\, d\mu = \int_A f d\mu \qquad \text{for $A \in \sigma(f)$}.$$

Definition. Say $f_n$ converges in ??? to a measurable function $f_n:\mathbb{X} \to \mathbb{Y}$ if $f_n$ converges to $f$ in measure and for all $\mu$-integrable functions $g:\mathbb{X} \to \mathbb{R}$, we have that $E(g | f_n)$ converges to $E(g | f)$ in measure.

Question. Does "converges in ???" have a formal name (for the mode of convergence or for the topology it induces on the space of measurable functions modulo a.e.-equivalence)? Is there any places or standard resources where it is used?


Question in probability notation

Let $X_n$ be a sequence of random variables taking values in a complete separable metric space $(\mathbb{X},d)$ (if it helps, assume $\mathbb{X}=\mathbb{R}$).

It is well-known that $X_n$ converges in probability to a random variable $X$ if $P\{d(X_n, X) > \varepsilon\} \to 0$ as $n \to \infty$ for all $\varepsilon > 0$.

Definition. Say $X_n$ converges in ??? to a random variable $X$ if $X_n$ converges to $X$ in measure and for all bounded continuous functions $g:\mathbb{X}^\infty \to \mathbb{R}$, we have that $E(g(X,X_1,X_2,\ldots) | X_n)$ converges to $E(g(X,X_1,X_2,\ldots) | X)$ in probability.

Question. Does "converges in ???" have a formal name (for the mode of convergence or for the topology it induces on the space of random variables modulo a.s.-equivalence)? Is there any places or standard resources where it is used?


Bonus Question

As I hinted to above, this approach also gives a notion of convergence of sigma-algebras (for a particular measure space $(\mathbb{X},\mu)$ as above). Fix a sequence of sub-sigma algebras $\mathcal{F}_n$ of the Borel sigma-algebra. Say that $\mathcal{F}_n$ converges to $\mathcal{F}$ if $E(g | \mathcal{F}_n)$ converges to $E(g | \mathcal{F})$ in measure for all $\mu$-integrable functions $g:\mathbb{X}\to\mathbb{R}$. (For example, by the Levy 0-1 law, if $\mathcal{F}_n$ is a filtration, then $\mathcal{F}_n$ converges to $\mathcal{F}_\infty$ in this sense.)

Question. Does this notion of convergence on sigma-algebras have a formal name (for the mode of convergence or for the topology it induces on the space of sigma-algebras modulo a.e.-equivalence)? Is there any places or standard resources where it is used?


Appendix 1

This mode of convergence is not the same as convergence in measure/probability. Here is a simple example. Let $U$ be a random variable uniformly distributed in $[0,1]$. Let $X_n = U/n$ and $X=0$. It is clear that $X_n$ converges to $X$ in probability, but it is not true that $X_n$ converges to $X$ in this new mode of convergence, since $E(X_1|X_n) = E(U|X_n) = U$, but $E(X_1|X) = E(U|X) = E(U) = 1/2$.


Appendix 2

It should be pointed out that the measure theory and probability theory definitions I gave differ in more than just notation. For one, the probability theory definition doesn't use the space $(\Omega,P)$ at all. However, if $\Omega$ is a completely separable measure space, $P$ is a Borel probability measure, and $X$ and $X_n$ are all Borel measurable, then the two definitions are equivalent. Here is a proof sketch.

Assume $E(h|X_n)$ converges in probability/measure to $E(h|X)$ for all $P$-integrable continuous $h:\Omega \to \mathbb{R}$. Then $g(X,X_1,X_2,\ldots)$ is $P$-integrable and therefore, $E(g(X,X_1,X_2,\ldots) | X_n)$ converges to $E(g(X,X_1,X_2,\ldots) | X)$ in probability.

Conversely assume $E(g(X,X_1,X_2,\ldots) | X_n)$ converges to $E(g(X,X_1,X_2,\ldots) | X)$ for all bounded continuous functions $g$. Let $f$ be a $P$-integrable function. Then using the tower property of conditional expectation we have that $E(f|X_n) = E(E(f|X,X_1,X_2 \ldots)|X_n)$. The function $E(f|X,X_1,X_2 \ldots)$ is $P$-integrable and therefore, there is a $P_(X,X_1,X_2,\ldots)$-integrable function $h:\mathbb{X}^\infty \to \mathbb{R}$ such that $E(f|X,X_1,X_2 \ldots) = h(X,X_1,X_2,\ldots)$ a.s.

So it remains to show that $E(h(X,X_1,X_2,\ldots) | X_n)$ converges to $E(h(X,X_1,X_2,\ldots) | X)$ in probability. The main idea is that since $h$ is integrable, we can approximate it with a continuous functions $g$ which are close in the $L_1$-norm. (The formal computation is left to the reader.)

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I'm not sure if this answers the bonus question, but topologies like that have been studied in mathematical economics to model private information of agents. Chapter 9 of the "Handbook of Applied Analysis" by Nikolaos S. Papageorgiou and Sophia Th. Kyritsi-Yiallourou contains a lot of material on the subject and you may find there what you need, it certainly studies the variant of your topology in which convergence in measure is replaced by $L_1$-convergence.

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  • $\begingroup$ I will have a look! By the way, for convergence of sigma-algebras, it doesn't matter if one uses convergence in measure or convergence in $L_1$, since it is enough to consider bounded functions $g:\mathbb{X}\to\mathbb{R}$. (For now I won't accept your answer since it doesn't answer my main question.) $\endgroup$ – Jason Rute Jan 10 '17 at 1:36
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I don't believe it is standard in probability theory to denote that mode of convergence in a specific manner. That being said, in the special case where the conditioning $\sigma$-fields are indexed and filtered by any set $I\subset \mathbb{R}$, there is convergence of $E(g|\mathcal{F_t})$, $t\in I$ along any increasing or decreasing sequence. In this case, one talks about $\textit{martingale convergence}$, or $\textit{reverse-martingale convergence}$, respectively.

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  • $\begingroup$ By the way, standard convention on these MathJax- and Markdown-powered sites prefers "martingale convergence or reverse-martingale convergence" (*martingale convergence* or *reverse-martingale convergence*) to the somewhat baroque "$\textit{martingale convergence}$ or $\textit{reverse-martingale convergence}$" ($\textit{martingale convergence}$ or $\textit{reverse-martingale convergence}$). $\endgroup$ – LSpice Jan 22 at 17:35

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