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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measure theory

# Too large to measure

# Slicing up compact sets

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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Show that if is a compact subset of , and if every intersection of with an hyperplane has measure zero in , then has measure zero.

*Note *This is valid for any measurable set in any -compact measure space by virtue of Tonelli’s theorem. The point is to prove this without using this. The fact that is compact will come in handy.