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For example what are the quasi-invariant probability measures on $\mathbb{R}$ with respect to translations?

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For your specific question about $\mathbb{R}$: from the Springer EOM

If $X$ is a topological homogeneous space with a continuous locally compact group of automorphisms $G$ ... and $B$ is the Borel $\sigma$-algebra ... then there [is a unique (up to measure equivalence) quasi-invariant measure].

For Euclidean spaces, all quasi-invariant (with respect to translations) measures are therefore equivalent to the Lebesgue measure. So all quasi-invariant probability measures are of the form $f~\mathrm{d}\mu$, where $\mathrm{d}\mu$ is the Lebesgue measure, and $f$ is an almost everywhere positive Lebesgue measurable function with $\int f~\mathrm{d}\mu = 1$.

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