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Given a measure space $(X,\Sigma,\mu)$, a measurable space $(Y,\Xi)$ and a measurable map $f\in\Sigma/\Xi$, the pushforward measure $\nu:=\mu\circ f^{-1}$ is given by $$ \nu[A] = \mu[f^{-1}(A)] $$ for any $A\in \Xi$. That is, $f$ pushes $\mu$ from $(X,\Sigma)$ to $(Y,\Xi)$. The notation $\mu\circ f^{-1}$ is often used in probability theory, but it is not really convenient. I saw also a notation with an asterisk such as $\nu = f^*\mu$ here and $\nu = f_*\mu$ here and here. Since I have no much experience with pushforwards and pullbacks in other fields, I am not sure which of the notations is the right one: with an asterisk on the top or bottom. I would be happy if somebody clarifies this.

P.S. If one considers the category of measurable maps and stochastic (conditional) kernels as arrows, $f$ and $\mu$ are both special cases of arrows so actually in that notation it appears that $$ \nu = f\circ \mu $$ which is very confusing taking into account the probabilistic variant $\nu = \mu\circ f^{-1}$.

I've found another notation $f_\#\mu$ here.

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SBF
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  • There is a general notational convention that a star upstairs denotes something contravariant, like a pull-back, an adjoint, and so on, i.e. something satisfying $(g\circ f)^\ast = f^\ast \circ g^\ast$, and a star downstairs means something covariant, like a push-forward, i.e., $(g \circ f)\ast = g\ast \circ f_\ast$. – Martin May 26 '13 at 17:42
  • @Martin: I see, so in the case of OP one shall use $f_*$, right? – SBF May 26 '13 at 18:20
  • Yes, that's what I would use and that's what I'm used to in connection with measures. However, I don't know what probabilists use, hence only a coment. // By the way: you look like Vyacheslav Tikhonov :-) – Martin May 26 '13 at 18:22
  • @Martin: that's indeed his photo :) I was looking for good photos of Shtirlitz, with no nazi uniform (not to be misunderstood), and that one appeared to me the best. Btw, have you seen notation $f_#\mu$ (I added a link in OP)? – SBF May 26 '13 at 18:42
  • It's a good choice :-) // I don't think I've seen $f_{#} \mu$ for measures, but in other contexts. I guess it would be an acceptable choice. In my view of the meaning of notation, the more important aspect than the symbol is whether it is a subscript or a superscript, partly for the reason explained in my first comment. – Martin May 26 '13 at 18:55
  • @Martin: I see :) // how do you know Tikhonov? – SBF May 26 '13 at 19:06
  • For some reason I don't remember I happened to come across the Isaev series and read a few of those novels. Out of curiosity, I watched a bit of the TV show. – Martin May 26 '13 at 19:15
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    What"s wrong with $\nu = \mu\circ f^{-1}$? It is after all the composition of the measure with the function $f^{-1}$ that maps sets to preimages. – Michael Greinecker May 27 '13 at 01:21
  • @MichaelGreinecker: it is not very convenient when you have to deal with compositions of functions $f\circ g\circ h$ and when you need to consider $\nu$ as a function of $\mu$. – SBF May 27 '13 at 08:15
  • I've seen all of the following been used: $\mu\circ f^{-1}$, $\mu_f$ and $f(\mu)$. However, I prefer using the first my self. – Stefan Hansen Jun 04 '13 at 08:09

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