# 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.

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.