# That’s because there’s no infinite dimensional analog to Lebesgue measure. Hilbert space allows for infinite dimensions and gauge groups… well… here https://arxiv.org/abs/1607.03591 – some nice proofs I found on why length and such just won’t work… but in my way of thinking without lengths, you can’t measure and if you can’t measure, you got nothing physical. Without anything physical, you have no physical observables.

That’s because there’s no infinite dimensional analog to Lebesgue measure.

Hilbert space allows for infinite dimensions and gauge groups… well… here https://arxiv.org/abs/1607.03591 – some nice proofs I found on why length and such just won’t work…

but in my way of thinking without lengths, you can’t measure and if you can’t measure, you got nothing physical. Without anything physical, you have no physical observables.
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Ah there

“A more intuitive idea why infinite-dimensional Lebesgue measure can’t exist comes from considering the effect of scaling. In R n , the measure of B (0 , 2) is 2 n times larger than B (0 , 1). When n = ∞ this suggests that one cannot get sensible numbers for the measures of balls.

There are nontrivial translation-invariant Borel measures on infinite-dimensional spaces: for instance, counting measure. But these measures are useless for analysis since they cannot say anything helpful about open sets.”

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