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)