Geometric stability theory for $\mu$-structures

We introduce a notion of $\mu$-structures which are certain locally compact group actions and prove some counterparts of results on Polish structures(introduced by Krupinski in \cite{Kru5}). Using the Haar measure of locally compact groups, we introduce an independence, called $\mu$-independence, in $\mu$-structures having good properties. With this independence notion, we develop geometric stability theory for $\mu$-structures. Then we see some structural theorems for compact groups which are $\mu$-structure. We also give examples of profinite structures where $\mu$-independence is different from $nm$-independence introduced by Krupinski for Polish structures.