  Let $F$ be a non-Archimedean locally compact field and $G$ a connected
reductive group defined over $F$. To any unipotent element $u$ in $G(F)$, we
have associated in [L] an $F$-stratum $\boldsymbol{\mathfrak{Y}}_{F,u}$ which
is a (possibly infinite) union of unipotent $G(F)$-orbits. We define here a
"canonical" non-zero positive $G(F)$-invariant Radon measure on
$\boldsymbol{\mathfrak{Y}}_{F,u}$. Under additional assumptions, we deduce the
convergence of the orbital integral associated to the $G(F)$-orbit of $u$. The
construction, valid in any characteristic, generalizes the one of Deligne-Ranga
Rao [RR] and also applies to nilpotent strata in $\textrm{Lie}(G)(F)$.
