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On the Wikipedia page on space-filling curves I see the following statement:

Wiener pointed out in The Fourier Integral and Certain of its Applications that space filling curves could be used to reduce Lebesgue integration in higher dimensions to Lebesgue integration in one dimension.

However as pointed out on this page at Is it true that a space-filling curve cannot be injective everywhere? spacefilling curves are not one-to-one mappings to $\mathbb{R}^2$. So is Wieners statement wrong?

What about the mappings discussed at Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$ . Could such a mapping be used to reduce higher dimensional integrals to lower dimensional ones, if space-filling curves don't work?

asmaier
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    Those maps aren't bijective, either, as the post you quoted admits: decimal expansion isn't unique in all points, $1.000\ldots=0.999\ldots$. Most ignore that, because the set of exceptional points has measure zero (it's countable, actually). So the reduction is possible, but not very practical: the space-filling curve isn't very regular, it's Hölder continuous, but nowhere differentiable, so numerically, those integrals aren't very nice. –  Sep 19 '17 at 19:24
  • If you are still interested, take a look at Hans Sagan's "Space-Filling Curves". – Red Banana May 08 '19 at 20:12

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