In general topology, one sometimes compares different topologies on the
same set. In particular, I am thinking of the various topologies on
function spaces, but there are several toplogies (mainly for examples,
it seems) even on the reals (e.g., the Sorgenfrey line).

In what I have seen of analysis (not much beyond first-year graduate
study), it seems that one compares measures on the same measurable space
(e.g., in the context of absolute continuity), but I have rarely seen
topics where one compares different sigma-fields on the same set. The
only example I can think of is conditional expectation in probability,
which is defined with respect to a sub-sigma-field of the underlying
probability space.

Are there any interesting and/or important examples where one compares
implication of different sigma-fields on the same set? I suspect one of
the issues here is that with unrelated sigma-fields, one cannot compare
measures.

[... = my original post]


Here's one (Blackwell's theorem): Let (Omega, F) be a countably
generated measurable space with the property that for each F-measurable
map f: Omega -> R, and each B in F, the image f(B) is an analytic subset
of R. Let G and H be sub sigma fields of F, H assumed to be countably
generated. Then G is contained in H if and only if each G-atom is a
union of H-atoms. [The G-atom (for example) containing x (a point of
Omega) is the intersection of all sets in G that contain x.]

Sorry for my ignorance, but what is an analytic subset? (I know what an

Corollary: A random variable X (aka, an F-measurable function from Omega
to R) is H-measurable if and only if X is constant on each H-atom.

ArtflDodgr wrote:

[... = my original post]


Here's one (Blackwell's theorem): Let (Omega, F) be a countably
generated measurable space with the property that for each F-measurable
map f: Omega -> R, and each B in F, the image f(B) is an analytic subset
of R. Let G and H be sub sigma fields of F, H assumed to be countably
generated. Then G is contained in H if and only if each G-atom is a
union of H-atoms. [The G-atom (for example) containing x (a point of
Omega) is the intersection of all sets in G that contain x.]

Sorry for my ignorance, but what is an analytic subset? (I know what an
analytic *function* is.)

Corollary: A random variable X (aka, an F-measurable function from Omega
to R) is H-measurable if and only if X is constant on each H-atom.