Infinite-dimensional Lebesgue Measure - Proof of The Theorem

Proof of The Theorem

Let X be an infinite-dimensional, separable Banach space equipped with a locally finite, translation-invariant measure μ. Using local finiteness, suppose that, for some δ > 0, the open ball B(δ) of radius δ has finite μ-measure. Since X is infinite-dimensional, there is an infinite sequence of pairwise disjoint open balls B_n(δ/4), n ∈ N, of radius δ/4, with all the smaller balls B_n(δ/4) contained within the larger ball B(δ). By translation-invariance, all of the smaller balls have the same measure; since the sum of these measures is finite, the smaller balls must all have μ-measure zero. Now, since X is a separable normed space, it is also a second-countable space and a Lindelöf space; hence, it can be covered by a countable collection of balls of radius δ/4; since each such ball has μ-measure zero, so must the whole space X, and so μ is the trivial measure.