Measure, Integration & Real Analysis · Problem 8C.15

Real and Complex Measures — Sheldon Axler

Mathematics — Measure Theory · no-translation-invariant-measure-infinite-dimensional

Problem

Suppose

V

is an infinite-dimensional Hilbert space. Prove that there does not exist a translation invariant measure on the Borel subsets of

V

that assigns positive but finite measure to each open ball in

V

. [A subset of

V

is called a Borel set if it is in the smallest

\sigma

-algebra containing all the open subsets of

V

. A measure

\mu

on the Borel subsets of

V

is called translation invariant if

\mu(f + E) = \mu(E)

for every

f \in V

and every Borel set

E

V