An irreducible partition of a space is a partition of that space into solid sets with a certain minimality property. Previously, these partitions were studied using the cup product in cohomology. This paper obtains similar results using the fundamental group instead. This allows the use of covering spaces to obtain information about irreducible partitions. This is then used to generalize Knudsen's construction of topological measures on the torus. We give examples of such measures that are invariant under Hamiltonian flows on certain symplectic manifolds.