# Is the Hausdorff metric on sub-$\sigma$-fields separable?

Let $$(X,\mu,\mathcal{F})$$ be a probability space. The paper Equiconvergence of Martingales by Edward Boylan introduced a pseudometric on sub-$$\sigma$$-fields (sub-$$\sigma$$-algebras) of $$\mathcal{F}$$ as follows:

$$\rho(\mathcal{G},\mathcal{H}) := \sup_{A\in \mathcal{G}} \inf_{B\in \mathcal{H}} \mu(A \triangle B) + \sup_{B\in \mathcal{H}} \inf_{A\in \mathcal{G}} \mu(A \triangle B)$$

where $$A \triangle B$$ is symmetric difference.

It seems to be called the Hausdorff pseudometric on $$\sigma$$-fields in later papers. (Does anyone know why?) Further, if we only consider a $$\mu$$-complete $$\sigma$$-fields then $$\rho$$ is a metric. Also, the paper shows $$\rho$$ is complete.

Is this metric $$\rho$$ separable---assuming, say, $$X=[0,1]$$ and $$\mu$$ is the Lebesgue measure?

My guess is that it is not, but I cannot off-hand come up with a witnessing set to show this. Considering the paper is 40 years old, I imagine this might be well-known. And if it is not separable, then my follow up question is this?

Is there a known separable, complete metric on the space of $$\mu$$-complete sub-$$\sigma$$-fields?

For reference, I found the following list $20of$20sigma-fields%22%7Csort:date/sci.math/iz249jCUvEU/rkKOei1NV5YJ" rel="nofollow noreferrer">online, compiled by Dave L. Renfro, of papers dealing with metrics on $$\sigma$$-fields (listed in Chronological order). I quickly looked though these papers and didn't find what I was looking for, but maybe I missed something.

1. Edward S. Boylan, "Equiconvergence of martingales",
Annals of Mathematical Statistics 42 (1971), 552-559. [MR 44 #7603; Zbl 218.60049]

2. Jacques Neveu, "Note on the tightness of the metric on the set of complete sub sigma-algebras of a probability space", Annals of Mathematical Statistics 43 (1972), 1369-1371. [MR 48 #5133; Zbl 241.60036]

3. Hirokichi Kudo, "A note on the strong convergence of sigma-algebras", Annals of Probability 2 (1974), 76-83. [MR 51 #6900; Zbl 275.60007]

4. Lothar Rogge, "Uniform inequalities for conditional expectations", Annals of Probability 2 (1974), 486-489. [MR 50 #14858; Zbl 285.28010]

5. Louis H. Blake, "Some further results concerning equiconvergence of martingales", Revue Roumaine de Mathématiques Pures et Appliquées 28 (1983), 927-932. [MR 86i:60130; Zbl 524.60029]

6. Hari G. Mukerjee, "Almost sure equiconvergence of conditional expectations", Annals of Probability 12 (1984), 733-741. [MR 86c:28012; Zbl 557.28001]

7. Beth Allen, "Convergence of sigma-fields and applications to mathematical economics", pp. 161-174 in Gerald Hammer and Diethard Pallaschke (editors), SELECTED TOPICS IN OPERATIONS RESEARCH AND MATHEMATICAL ECONOMICS (Proceedings, Karlsruhe, West Germany, 22-25 August 1983), Lecture Notes in Economics and Mathematical Systems #226, Springer-Verlag, 1984. [MR 86f:90029; Zbl 547.28001]

8. Dieter Landers and Lothar Rogge, "An inequality for the Hausdorff-metric of sigma-fields", Annals of Probability 14 (1986), 724-730. [MR 87h:60006; Zbl 597.60003]

9. Abdallah M. Al-Rashed, "On countable unions of sigma algebras", Journal of Karachi Mathematical Association 8 (1986), 57-63. [MR 88f:28001; Zbl 639.28001]

10. Maxwell B. Stinchcombe, "A further note on Bayesian information topologies", Journal of Mathematical Economics 22 (1993), 189-193. [MR 93k:60011; Zbl 773.90016]

11. Timothy Van Zandt, "The Hausdorff metric of sigma-fields and the value of information", Annals of Probability 21 (1993), 161-167. [MR 94d:62012; Zbl 777.62007]

12. Xikui Wang, "Completeness of the set of sub-sigma-algebras", International Journal of Mathematics and Mathematical Sciences 16 (1993), 511-514. [MR 94f:28002; Zbl 782.28001]

13. Zvi Artstein, "Compact convergence of sigma-fields and relaxed conditional expectation", Probability Theory and Related Fields [= Zeitschrift für Wahrscheinlichkeits- theorie] 120 (2001), 369-394. [MR 2002g:28003; Zbl 992.28001]

• It is called the Hausdorff pseudometric because it is an instance of Haudsorff's construction starting with pseudometric $\mu(A \triangle B)$. In general, Hausdorff's construction starts with a pseudometric and constructs a new pseucometric on subsets of the original space. See here en.wikipedia.org/wiki/Hausdorff_distance . Nov 3, 2011 at 0:25
• As a first step, what is the cardinality of the set of complete $\sigma$-algebras? If it isn't $2^{\aleph_0}$, then that's certainly an obstruction. Nov 3, 2011 at 4:52
• @Gerald Edgar, thanks I thought it might be something like that. Nov 3, 2011 at 14:12
• @Nate Eldredge, I think it is size continuum as follows: If $\rho(\mathcal{G},\mathcal{H})\neq 0$, then $f \mapsto E[f \mid \mathcal{G}]$ and $f \mapsto E[f \mid \mathcal{H}]$ are different operators. But in $L^2$ these operators are continuous linear transformations of which there are only continuum many (correct?). Nov 3, 2011 at 14:18
• Maybe someone can fix this? Apparently the substitution principle doesn't hold for URL exchanges. Jan 16, 2020 at 13:36

Take a sequence $A_n$ of independent sets of measure $1/2$. Given two different subsets $B$ and $C$ of natural numbers, suppose WLOG that there is an $n$ in $B\sim C$. Now $\mu(A_n\Delta A) = 1/2$ for all sets $A$ which are independent of $A_n$, so the distance from the sigma algebra generated by $(A_n)_{n\in B}$ to the sigma algebra generated by $(A_n)_{n\in C}$ is at least $1/2$. This shows that the density character of your space is at least the continuum.
• Thanks! I assume $B\sim C$ is set subtraction. I feel a bit silly for not noticing this construction. Also, by my comment to Nate above, the space has size continuum, so the density character is exactly the continuum. Nov 3, 2011 at 14:30
For each $n\in\mathbb{N}$ take all sub-algebras of the finite sigma-algebra generated by intervals of the form $[i/2^n,(i+1)/2^n)$, $i=0,\ldots,2^n-1$.
• Unfortunately not. (I had thought this at first too.) By Bill Johnson's answer, every pair in your sequence has distance 1/2. (So this sequence doesn't even converge in this metric to the Borel $\sigma$-algebra which it generates.) A similar example is in the original paper. (I just didn't see at the time how to extend it to a collection of size continuum as Bill did.) Nov 3, 2011 at 14:39