Let be an infinite dimensional normed space. There is no non-trivial translation invariant Borel measure on which is finite on open balls.

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# Too large to measure

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Let be an infinite dimensional normed space. There is no non-trivial translation invariant Borel measure on which is finite on open balls.

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Using Riesz’s lemma, we may construct a sequence such that and for . Then, the balls are all disjoint and contained in . If the measure is finite on open balls, it must follow, by translation-invariance and -additivity, that the measure of any ball of radius is zero, which implies that the measure is trivial.

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