On the classification of groups of orientation-preserving homeomorphisms of $\mathbb R$. III. $\omega$-projectively invariant measures

General groups of orientation-preserving homeomorphisms of $\mathbb R$ are investigated. A series of metric invariants are defined for such groups: $\omega$-projectively invariant measures, where $\omega$ is a cardinal number. A theorem on the existence of an $\omega$-projectively invariant measure is formulated, which is a natural generalization of the Bogolyubov–Krylov theorem on the existence of an invariant measure for a circle homeomorphism. For groups with an $\omega$-projectively invariant measure “obstructions” to the existence of a 1-projectively invariant measure are analysed. The approach is based on the study of the topological structure of the set of all fixed points of the elements of the group, the orbits of points in the line, minimal sets, and the combinatorial properties of groups.