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In this paper, we study the volume of algebraically integrable foliations and locally stable families. We show that, for any canonical algebraically integrable foliation, its volume belongs to a discrete set depending only on its rank and the volume of its general leaves. In particular, if the foliation is of general type, then its volume has a positive lower bound depending only on its rank and the volume of its general leaves. This implies some special cases of a question posed by Cascini, Hacon, and Langer. As a consequence, we show that the relative volume of a stable family with a normal generic fiber belongs to a discrete set if the dimension and the volume of its general fibers are bounded. Log versions of the aforementioned theorems are also provided and proved., Comment: 24 pages

We study the structures of $\epsilon$-lc Calabi--Yau pairs. We show that such pairs possess a finite cover that is isomorphic in codimension 1 with a fiber product of a Calabi--Yau variety and a rationally connected variety. Moreover, the degree of the cover depends only on $\epsilon$ and the dimension. As a corollary, we show that the index of 4-dimensional non-canonical Calabi--Yau variety is bounded. We also prove a boundedness result on polarized $\epsilon$-lc Calabi--Yau pairs., Comment: 32 pages, comments welcome!