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I have some questions, but not sure if they are meaningful:

  1. Suppose $X$ and $Y$ are two arbitrary measurable spaces. Does there exist a measurable mapping from $X$ to $Y$?
  2. Suppose $X$ and $Y$ are two arbitrary measure spaces. Does there exist a measure-preserving mapping from $X$ to $Y$?
  3. Suppose $X$ and $Y$ are two arbitrary topological spaces. Does there exist a continuous mapping from $X$ to $Y$?

They are similar in this way:

if $X$ and $Y$ are two arbitrary sets with some type of structure, does there exist a structure-preserving mapping from $X$ to $Y$?

Hope to see if there can be some insights.

Thanks and regards!

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    Every constant map $f:X\to Y$ is measurable and continuous (wrt. to every $\sigma$-algebra over $Y$ and every topology over $Y$, respectively), so answers to the first and third questions are positive. The answer to the second question is negative -- consider the measure space $Y$ with the zero measure and $X$ with a nonzero measure.2011-11-25
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    Thanks! How about non constant mapping?2011-11-25
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    Re measurability, all I can think is that if the map is differentiable, then the measure is preserved precisely when the determinant of the Jacobian is 1. since a mapping f between abstract measurable spaces $(X,\mu,\sigma), (Y,\phi, \sigma'$ is measurable if/when $f^{-1}(U)$=V, where U is in \sigma and V is in sigma', you can always construct a measurable map f between them, by demanding than an element in sigma' be sent to an element in sigma under $f^{-1}$. Same goes for topological2011-11-25
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    (cont.)spaces A,B, whereyou can just construct g:A-->B , so that $g^{-1}(W)=Z$ , for any W open in Y, and some Z open in X. I think this is part of what pointless topology is about. Is that your question?2011-11-25
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    @Tim: if $X$ is connected with cardinality greater than 1 and $Y$ is discrete, then there is no continuous map from $X$ to $Y$ other than a constant one.2011-11-25
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    @DamianSobota: Thanks! Why is that?2011-11-25
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    @gary: Thanks! (1) Do you mean point set topology instead of pointless topology? (2) How do you "demanding than an element in sigma' be sent to an element in sigma under $f^{-1}$"?2011-11-25
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    @Tim: by pointless topology, I am referring to:http://en.wikipedia.org/wiki/Pointless_topology . (2): We can define f so that the set of points in the element W in the sigma-algebra of the target is sent to the set Z in the sigma algebra of the initial space. This defines a map between the two measure spaces.2011-11-25
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    For your boxed question, the answer really depends on the type of structure. For example, generally you cannot guarantee the existence of a field homomorphism between two arbitrary fields (since such a map must be an embedding, if you include $1$ in your structure). You cannot guarantee maps between partially ordered sets, if you require your structure to respect strict ordering. Etc.2011-11-25
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    @ArturoMagidin: Thanks for mentioning about algebraic structures!2011-11-25
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    @gary: Thanks! Still don't quite understand "We can define f so that the set of points in the element W in the sigma-algebra of the target is sent to the set Z in the sigma algebra of the initial space. "2011-11-25
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    @Tim: I need to be out for a while, but I'll be back soon, and will expand on it then.2011-11-25
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    @Tim : I'm finally back; do you want more of a followup?2011-11-26
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    @gary: Yes, please. Maybe you can consider to turn your previous comments and future comments into a reply? Thanks!2011-11-26

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