(Plante) A codimension one minimal foliation has at most one invariant measure
This is true. Pick a complete transversal. If an invariant measure exists it yields a Borel probability without atoms and with full support on the transversal. The holonomy transformations preserve this measure and are therefore equicontinuous. Uniqueness of the invariant measure then follows from Shigenori Matsumoto's article "The unique ergodicity of equicontinuous laminations" (2010, Hokkaido Mathematical Journal).