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Wednesday, September 18, 2019

**Abstract:** Let $\mathrm{MALG}(X)$ denote the measure algebra of a standard probability space $(X,\mu)$. A measure preserving action of a Polish group $G$ on $\mathrm{MALG}(X)$ is called *whirly* if for any $A, B$ in $\mathrm{MALG}(X)$ with positive measure and for any open neighborhood $U$ of the identity of $G$ there exists $g\in U$ so that $(gA)\cap B$ has positive measure. We follow a paper of Glasner–Tsirelson–Weiss to show that if $G$ is certain type of Polish group, namely a *Lévy* group, then any non-trivial Borel action on $\mathrm{MALG}(X)$ is whirly. We also show that the Polish group $L_0(\mathbb{T})$ of all measurable functions $[0,1] \to \mathbb{T}$ is Lévy using a suitable concentration inequality.

This is a follow up to the lecture of D. Ihli on 09/11/2019.