Why do all measure theory textbooks present the concept of push-forward measure, but never the concept of pull-back measure? Doesn't the latter exist?


It's true that the naive treatment of such a concept would sometimes lead to contradictions. For instance, let $p:\mathbb{R}^2\rightarrow\mathbb{R}$ be given by $p(x,y)=x$, $\lambda$ be the Lebesgue measure on $\mathbb{R}$, and $A=[0,1]\times[0,1]$. If one decomposes $A=A\_1 \cup A\_2$ where $A\_1=[0,1]\times[0,\frac{1}{2}]$ and $A\_2=[0,1]\times(\frac{1}{2},1]$, then $(p^\* \lambda)(A)=1$ while $(p^\* \lambda)(A\_1)+(p^\* \lambda)(A\_2)=2$, showing that the naive definition of the pull-back does not lead to a measure.


Similarily, if $i:\mathbb{R}\rightarrow\mathbb{R}^2$ is given by $i(x)=(x,0)$, then the pull-back of the Lebesgue measure on $\mathbb{R}^2$ would be 0.


Given the two situations presented above, can one define a fruitful concept of pull-back measure?



Pullback measures
To define pullbacks of measures we need some additional data,
because otherwise one would be able
to obtain a canonical measure on an arbitrary measurable space M
by pulling back the canonical measure on the point along the unique map M→pt.


One natural choice for such additional data is
a choice of measure on each fiber of the map f: M→N.
Using such a fiberwise measure one can define the pullback measure f\*μ on M
given a measure μ on N as follows:
to integrate a function h on M with respect to f\*μ we integrate h fiberwise with respect
to the fiberwise measure on M and then we integrate
the resulting function on N with respect to μ.


To define pullbacks of complex valued measures
we have to require that the fiberwise measure
(which can now also be complex valued) is fiberwise finite
and its norm (total variation) is uniformly bounded
with respect to N.


To ensure that pullbacks of probability measures
are again probability measures we have to require
that fiberwise measures are probability measures.


All of this can be done in greater generality
for arbitrary noncommutative measurable spaces
(i.e., von Neumann algebras) and arbitrary L\_p-spaces
(instead of just L\_1-spaces, i.e., measures),
where p is an arbitrary complex number,
as explained in this answer:
[Is there an introduction to probability theory from a structuralist/categorical perspective?](https://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p/20820#20820)


