Do $C([0,1];\mathbb{R}^2)$ and $C([0,1];\mathbb{R}^3)$ with the respective Borel sigma algebras under the uniform topology differ as measure spaces?
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2026-04-13 10:13:18.1776075198
Do $C([0,1];\mathbb{R}^2)$ and $C([0,1];\mathbb{R}^3)$ differ as measure spaces?
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$C([0,1],\mathbb{R}^2)$ and $C([0,1],\mathbb{R}^3)$ are examples of Polish spaces, i.e. separable completely metrizable topological spaces. It's a general fact that for any uncountable Polish space $X$, there is a bimeasurable bijection from $X$ with its Borel $\sigma$-algebra to $[0,1]$ with its Borel $\sigma$-algebra.