Measurable Functions on Pulled-Back Measurable Spaces
We start today with a possibly surprising result; pulling back a -ring puts significant restrictions on measurable functions. If
is a function from a set into a measurable space
, and if
is measurable with respect to the
-ring
on
, then
whenever
.
To see this fix a point , and let
be a measurable set containing
. Its preimage
is then a measurable set containing
. We can also define the level set
, which is a measurable set since
is a measurable function. Thus the intersection
is measurable. That is, it’s in , and so there exists some measurable
so that
is this intersection. Clearly
, and so
is as well, by assumption. But then
, and we conclude that
.
From this result follows another interesting property. If is a mapping from a set
onto a measurable space
, and if
is a measurable function, then there is a unique measurable function
so that
. That is, any function that is measurable with respect to a measurable structure pulled back along a surjection factors uniquely through the surjection.
Indeed, since is surjective, for every
we have some
so that
. Then we define
, so that
, as desired. There is no ambiguity about the choice of which preimage
of
to use, since the above result shows that any other choice would lead to the same value of
. What’s not immediately apparent is that
is itself measurable. But given a set
we can consider its preimage
, and the preimage of this set:
which is measurable since is a measurable function. But then this set must be the preimage of some measurable subset of
, which shows that the preimage
is measurable.
It should be noted that this doesn’t quite work out for functions that are not surjective, because we cannot uniquely determine
if
has no preimage under
.