Consider a surjective map $f:(X, \sigma(f)) \to (Y, \mathcal{Y})$, if a measure $\nu$ is given in $(Y, \mathcal{Y})$ the pullback $\nu(f(\cdot))$ is a measure on $(X, \sigma(f))$, similarly if $\mu$ is given on $(X, \sigma(f))$ the push-forward $\mu(f^{-1}(\cdot))$ is a measure on $(Y, \mathcal{Y}).$

If the map $f$ is not a bijection there is no sense to talk about isomorphism between the probability spaces $(X, \sigma(f), \nu(f(\cdot)))$ and $(Y, \mathcal{Y}, \nu)$, at least not in the sense described in https://en.wikipedia.org/wiki/Standard_probability_space#Isomorphism

But there is still something to rescue since the push-forward of $\nu(f(\cdot))$ is again $\nu$, and if $\mu$ is a measure on $(X, \sigma(f))$ the push of $\mu$, $\mu(f^{-1}(\cdot))$ has as pull-back again $\mu$.

At the end it depends on the fact that for any set $A \subset Y$ we have $f(f^{-1}(A)) = A$ and for the generator sets in $\sigma(f)$ we have $f^{-1}(f(B)) = B$, if $B = f^{-1}(A)$ then $f^{-1}(f(B)) = f^{-1}(f(f^{-1}(A))) = f^{-1}(A) = B$.

So even if we have no injectivity the two spaces are extremely connected, in some sense this looks a similar construction as the isomorphism theorem in group theory.

What exactly is the term used to describe the relation between the spaces in this case? or the part of measure theory associated with this construction?


  • $\begingroup$ What is the pull-back here? $\endgroup$ – Michael Greinecker Dec 20 '16 at 23:04
  • $\begingroup$ @MichaelGreinecker When I write $\mu(f(\cdot))$ I mean the probability measure $\nu$ on $(X, \sigma(f))$ that for generators $B = f^{-1}(A)$ gives the probability $\nu(B) = \mu(f(B))$, since $f(f^{-1}(A)) = A$ for surjective maps, this means $\mu(f(B)) = \mu(f(f^{-1}(A))) = \mu(A)$. $\endgroup$ – Jorge E. Cardona Dec 20 '16 at 23:52
  • $\begingroup$ There are lots of measures on X that push forward to any given measure on Y. One natural way to pull back is to use conditional expectation. $\endgroup$ – Anthony Quas Dec 21 '16 at 4:10
  • $\begingroup$ @AnthonyQuas I am aware that there are plenty of them before the conditioning (projection) but those are measure on a bigger sigma algebra no? I am looking for measures on $\sigma(f)$ which may come from a conditioning of different measures to $\sigma(f)$ but once conditioned I am expecting it to be unique. $\endgroup$ – Jorge E. Cardona Dec 21 '16 at 10:37

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