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 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.
Edward S. Boylan, "Equiconvergence of martingales",
Annals of Mathematical Statistics 42
(1971), 552-559. [MR 44 #7603; Zbl 218.60049]
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]
Hirokichi Kudo, "A note on the strong convergence of
sigma-algebras", Annals of Probability 2 (1974), 76-83.
[MR 51 #6900; Zbl 275.60007]
Lothar Rogge, "Uniform inequalities for conditional
expectations", Annals of Probability 2 (1974), 486-489.
[MR 50 #14858; Zbl 285.28010]
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]
Hari G. Mukerjee, "Almost sure equiconvergence of
conditional expectations", Annals of Probability 12
(1984), 733-741. [MR 86c:28012; Zbl 557.28001]
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]
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]
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]
Maxwell B. Stinchcombe, "A further note on Bayesian
information topologies", Journal of Mathematical Economics
22 (1993), 189-193. [MR 93k:60011; Zbl 773.90016]
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]
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]
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]
