I will specify the question. In a conference by Arnold (on continued fractions, you can find it on YouTube) he says that the fact that we consider preimages when defining measure preserving maps is something deep, which links to controvariant and covariant concepts in mathematics. What does this mean? I know the definition of measure preserving map, and I know Ergodic Theory and all the consequences, but is there something more deep about why we have to take the preimage and not the image when defining measure preserving functions?
We have a similar phenomena in topology and measure theory, when we define continuos and measurable maps: to say that a map preserve some kind of structure, we ask something involving the preimage. Is there something deep also in those definitions? In topology for example we have that the continuous image of a compact or of a connected is compact or connected, but the preimage of an open or a closed is an open or a closed (in topology we can define open and closed maps, but they are less used that continuous maps)
I have never asked myself this question before because in topology the definition of continuous map arises very easily from the epsilon-delta definition in metric space, and in measure theory the definition of measurable map it works well with integrals. But in Ergodic Theory if I were asked to chose the definition of measure preserving I would have said that the measure of the image is equale to the one of the starting set.
Are there similar example in Maths of this phenomena?
In short because for two subset$A,B$ of a space $X$, $f^{-1}(A \cap B)=f^{-1}(A) \cap f^{-1}(B)$ but there are case where $f(A \cap B) \ne f(A) \cap f(B)$ (it is a good exercice to show the first claim and find counterexample of the secound).
So the preimage of a $\sigma$-algebra is a $\sigma$-algebra but not the image.