A necessary and sufficient condition for a measure which is quasi-invariant under a transformation to be equivalent to an invariant probability measure is that the transformation cannot (in a measure theoretic sense) compress the space.
Hopf's Theorem
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Cite this as:
Weisstein, Eric W. "Hopf's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HopfsTheorem.html