  The von Neumann-Day problem asks whether every non-amenable group contains a
non-abelian free group. It was answered in the negative by Ol'shanskii in the
1980s. The measurable version (formulated by Gaboriau-Lyons) asks whether every
non-amenable measured equivalence relation contains a non-amenable treeable
subequivalence relation. This paper obtains a positive answer in the case of
arbitrary Bernoulli shifts over a non-amenable group, extending work of
Gaboriau-Lyons. The proof uses an approximation to the random interlacement
process by random multistep of geometrically-killed random walk paths. There
are two applications: (1) the Gaboriau-Lyons problem for actions with positive
Rokhlin entropy admits a positive solution, (2) for any non-amenable group, all
Bernoulli shifts factor onto each other.
