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I'm reading a text on measure-theoretic probability theory, and in a discussion of standard measurable spaces, the text mentions that $\mathbb{R}^n$ and $\mathbb{R}^\infty$, together with the Borel $\sigma$-algebras, are standard measurable spaces. Since this isn't a book on measure theory, this result is not proved anywhere in the text, so I'm wondering if there's an easy way to see that these two measurable spaces are indeed standard.

For reference, the definition of a standard measurable space given in the text is a measurable space that is isomorphic to a Borel subset of $\mathbb{R}$, together with the associated Borel $\sigma$-algebra.

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    $\begingroup$ Two Borel sets of Polish spaces are Borel isomorphic, iff they have the same cardinality. The proof is quite long, and you can find it on some descriptive set theory textbook, or Probability measures on metric spaces, Parthasarathy(1967). $\endgroup$ – Metta World Peace Aug 14 '13 at 21:53

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