Pulling Back and Pushing Forward Structure
Remember that we defined measurable functions in terms of inverse images, like we did for topological spaces. So it should be no surprise that we move a lot of measurable structure around between spaces by “pulling back” or “pushing forward”.
First of all, let’s say that is a measurable space and consider a function
. We can always make
into a measurable function by pulling back the
-ring
. For each measurable subset
we define the preimage
as usual, and define the pullback
to be the collection of subsets of
of the form
for
. Taking preimages commutes with arbitrary setwise unions and setwise differences, and
, and so
is itself a
-ring. Every point
gives us a point
, and every point
is contained in some measurable set
. Thus
is contained in the set
, and so we find that
is a measurable space. Clearly,
contains the preimage of every measurable set
, and so
is measurable.
Measures, on the other hand, go the other way. Say that is a measure space and
is a measurable function between measurable spaces, then we can define a new measure
on
by “pushing forward” the measure
. Given a measurable set
, we know that its preimage
is also measurable, and so we can define
. It should be clear that this satisfies the definition of a measure. We’ll write
for this measure.
If is a measurable function, and if
is a measure on
, then we have the equality
in the sense that if either integral exists, then the other one does too, and their values are equal. As usual, it is sufficient to prove this for the case of for a measurable set
. Linear combinations will extend it to simple functions, the monotone convergence theorem extends to non-negative measurable functions, and general functions can be decomposed into positive and negative parts.
Now, if is the characteristic function of
, then
if
— that is, if
— and
otherwise. That is,
. We can then calculate
As a particular case, applying the previous result to the function shows us that
We can go back and forth between either side of this equation by the formal substitution .
Finally, we can combine this with the Radon-Nikodym theorem. If is a measurable function from a measure space
to a totally
-finite measure space
so that the pushed-forward measure
is absolutely continuous with respect to
. Then we can select a non-negative measurable function
so that
again, in the sense that if one of these integrals exists then so does the other, and their values are equal. The function plays the role of the absolute value of the Jacobian determinant.