Complex vector measure and integral over manifolds with locally finite variations

Abstract

It is well known that one can integrate any compactly supported continuous complex differential n-form over real-n-dimensional C1-manifolds in Cm (m n). For n = 1 the integral may be defined over any locally rectifiable curve. Another generalization is the theory of currents (linear functionals on the space of compactly supported C∞-differential forms). The theme of the article is integration of measurable complex differential (n, 0)-forms (without d¯zj) over real-n-dimensional C0-manifolds in Cm with locally finite n-dimensional variations (a generalization of locally rectifiable curves to dimension n > 1). The last result states that a real-n-dimensional manifold, C1-embedded in Cm, has locally finite variations and the integral of measurable complex differential (n, 0)-form determined in the article may be calculated by well known formula. Refs 5.