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 $f_j$.

中文翻译：

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Alexandrov 空间收缩序列的纤维化定理

假设一个序列${米}_j$亚历山德罗夫空间坍缩成一个空间$X$只有弱奇点。山口制作了一张地图$F_j:{米}_j\to X$称为几乎 Lipschitz 淹没大$j$. 我们证明如果${米}_j$方向空间的体积有一个统一的正下界，与奇点的弱点相比足够大$X$， 然后$F_j$是局部平凡的纤维化。此外，我们展示了一些关于内在度量和纤维体积的属性$F_j$.