Space of fuzzy measures and representation of Choquet integral

The space of fuzzy measures with topology introduced by the Choquet integral is considered. We show that the subspace of fuzzy measures which is less than and equal to 1 is compact. The space of non-negative fuzzy measures is a locally convex space. Applying the theorems mentioned above, we obtain the additive representation of the Choquet integral. The similar theorems are shown in various contexts. We show that these theorems are partially equivalent, but they are not perfectly equivalent. The differences among them are stated in the forms of the table