  Suppose a sequence $M_j$ of Alexandrov spaces collapses to a space $X$ with
only weak singularities. Yamaguchi constructed a map $f_j:M_j\to X$ called an
almost Lipschitz submersion for large $j$. We prove that if $M_j$ has a uniform
positive lower bound for the volumes of spaces of directions, which is
sufficiently large compared to the weakness of singularities of $X$, then $f_j$
is a locally trivial fibration. Moreover, we show some properties on the
intrinsic metric and the volume of the fibers of $f_j$.
