To establish the relationship between conditional information entropy defined on boundary region and attribute reduction, it was proved that the conditional information entropy defined on discourse of universe was the same as the one on boundary region. It means that the representation of information entropy reduction can be obtained by conditional information entropy defined on boundary region. By strictly convex function and Jensen inequality, its properties were discussed. To remain conditional information entropy defined on boundary region unchanged, the sufficient and necessary condition was presented. To get the representation of positive region reduction by conditional information entropy defined on boundary region, its sufficient and necessary condition was given, so as to get the judgment approach for positive region reduction from the view of conditional information entropy on boundary region. It is the generalization of similar method for consistent decision information system. Finally, a numerical example was designed to show how to use the conditional information entropy defined on boundary region to compute the positive region or conditional information entropy reductions.