What kind of measure is there on the Borel algebra (or Baire algebra) of the Cantor space $2^{\omega_1}$?
Is it the Haar measure? If so, why is the Lebesgue measure not good enough?
2026-03-27 03:43:17.1774582997
Measure of the Cantor space $2^{\omega_1}$
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A summary of a few basic properties:
No measure is related to the Haar measure because it is not a group.
The usual topology is the Tychonoff topology and so you induce a measure by taking any measure on disjoint sets generating the $\sigma$-algebra (some people call them cylinders).
Depending on the construction one of those measures will be the Hausdorff measure of the right Hausdorff dimension.
The Lebesgue measure is zero and so it is not very useful.