# What mode of convergence is this?

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.)

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.
• 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.) Jan 10, 2017 at 1:36
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.
• 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 $\textit{reverse-martingale convergence}$). Jan 22, 2019 at 17:35