Point realization of Boolean actions of countable inductive limits of locally compact groups

Анотація

Let  $G$ be a CILLC-group, i.e., the inductive limit of an increasing sequence of its closed locally compact subgroups. Every nonsingular action of $G$ on a measure space $(X, \mathcal{B},\mu)$ generates a continuous action of  $G$ on the underlying Boolean $\sigma$-algebra $\mathcal{M}[\mu]=\mathcal{B}/I_\mu$, where $I_\mu$  is the ideal of $\mu$-null subsets. It is known that the converse is true for any locally compact  $G$: every abstract Boolean  $G$-space is associated with some Borel nonsingular action of  $G$. In the present work this assertion is generalized to arbitrary CILLC-groups. In addition, we conctruct a free measure preserving action of  $G$ on a standard probability space.