On a large class of infinite trees
T
T
, we prove the existence of harmonic functions
h
h
, with respect to suitable transient transition operators
P
P
, that satisfy the following universal property:
h
h
is the Poisson transform of a martingale on the end-point boundary
Ω
\Omega
of
T
T
(equipped with the harmonic measure induced by
P
P
) such that, for every measurable function
f
f
on
Ω
\Omega
, it contains a subsequence converging to
f
f
in measure. Moreover, the martingale visits every open set of measurable functions with positive lower density.