Wednesday, September 21, 2016

real analysis - bijective measurable map existence


Does there exist bijective measurable maps between $\mathbb{R}$ and $\mathbb{R}^n$?


If so, could you give me an example of that?


Thank you.


Answer



Polish space is a topological space that is isomorphic to a complete separable metric space, for example $\Bbb R^n$ for any $n\in \Bbb N$. For the proof of the following fact, see e.g. here.



Any uncountable Polish space is Borel isomorphic (there exists a bimeasurable bijection) to the space of real numbers $\Bbb R$ with standard topology.




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