Talk:Pushforward measure

Thanks

Thanks for elaborating this page. I have relinked to here as much as possible. Now I will take it off my watch list. Good luck! Geometry guy 00:17, 12 February 2007 (UTC)

Attention needed to the definition/examples? [Resolved]

The first example seems to state that the measure of an arc $A$ of the circle is equal to the measure of $f^{-1}(A)$ on the real line, where $f:\mathbb {R} \rightarrow C$ is the wrap-around function. But $f^{-1}(A)$ has measure $\infty$.

Should the correct definition define $\mu (A):=\inf _{f(B)=A}\mu (B)$? Am I missing something?

69.81.71.60 (talk) 11:54, 28 June 2017 (UTC)

No, why? It is written "Let λ also denote the restriction of Lebesgue measure to the interval [0, 2π)". Also f is defined on [0, 2π). Not infinity. Boris Tsirelson (talk) 18:47, 28 June 2017 (UTC)

I see now. Thank you! Norbornene (talk) 13:29, 9 July 2017 (UTC)

"Random variables are pushforward measures"

As far as I can see, the following statement is false:

Random variables are pushforward measures

A r.v. defines a pushforward measure, but there is not one-to-one identification. For example, i.i.d r.v.'s $Y_{i}:X_{1}\to X_{2}$ define the same pushforward measure $Y_{*}P$, although they are clearly distinct mappings from the probability space $(X_{1},\Sigma _{1},P)$ to a measurable space $(X_{2},\Sigma _{2})$. AVM2019 (talk) 12:50, 19 May 2022 (UTC)