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

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.