Let F be a non-discrete non-Archimedean locally compact field and OF the ring of integers in F. The main results of this paper are Theorem 1.2 that classifies ergodic probability measures on the space Mat(N,F)
Nearby in the stack
of infinite matrices with enties in
F
with respect to the natural action of the group
GL(∞,OF)×GL(∞,OF)
and Theorem 1.6 that, for non-dyadic
F
, classifies ergodic probability measures on the space
Sym(N,F)
of infinite symmetric matrices with respect to the natural action of the group
GL(∞,OF)
.
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