In [Theorem 2.7 in the following notes](http://www.math.nus.edu.sg/~matsr/ProbII/Lec1.pdf), we seem to assume the following statement.



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> Let $(\Omega,\mathcal F)$ be a Polish space, and $A\in\mathcal F$ an uncountable set. Then there exists a bijection $\phi:(A,\mathcal F |\_A)\to([0,1],\mathcal B([0,1]))$ such that $\phi$ and $\phi^{-1}$ are both measurable.
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Now, in the book *Probability Measures on Metric Spaces* by Parthasarathy et al, Theorem I.2.12 states the following:



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> Let $X\_1,X\_2$ be separable complete metric spaces, and $E\_i\subset X\_i$ Borel subsets, for $i=1,2$. Then $E\_1$ and $E\_2$ are isomorphic (in the sense described above) iff they have the same cardinality.
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But do we know that any uncountable Borel-measurable $A\subset\Omega$ where $A$ is polish necessarily has cardinality at least that of the continuum?



Measurably-isomorphic subsets of polish spaces and the continuum hypothesis
Every Borel subset (and in fact every analytic subset) of a Polish space either is countable or has a perfect subset. In particular, an uncountable Borel subset in a Polish space has the cardinality of the continuum. This can be found, for example, in Moschovakis's book "Descriptive Set Theory" as Corollary 2C.3. The result apparently goes back to Suslin and Lusin in 1917.


