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).
Comments
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).