“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.”