Too large to measure

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

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Too large to measure

One thought on “Too large to measure

  1. Using Riesz’s lemma, we may construct a sequence (x_n)\in X such that \Vert x_i\Vert=1 and \Vert x_i-x_j\Vert > 1/2 for i\neq j. Then, the balls B_{1/2}(x_i) are all disjoint and contained in B_{3/2}(0). If the measure is finite on open balls, it must follow, by translation-invariance and \sigma-additivity, that the measure of any ball of radius 1/2 is zero, which implies that the measure is trivial.

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