Volume Form - Relation To Measures

Relation To Measures

Given a volume form ω on an oriented manifold, the density |ω| is a volume pseudo-form on the nonoriented manifold obtained by forgetting the orientation. Densities may also be defined more generally on non-orientable manifolds.

Any volume pseudo-form ω (and therefore also any volume form) defines a measure on the Borel sets by

The difference is that while a measure can be integrated over a (Borel) subset, a volume form can only be integrated over an oriented cell. In single variable calculus, writing considers as a volume form, not simply a measure, and indicates "integrate over the cell with the opposite orientation, sometimes denoted ".

Further, general measures need not be continuous or smooth: they need not be defined by a volume form, or more formally, their Radon–Nikodym derivative with respect to a given volume form needn't be absolutely continuous.