Rectifiable Measures in R n and Existence of Principal Values for Singular Integrals

A Constructive Minimal Integral which Includes Lebesgue Integrable Functions and Derivatives

In this paper we provide a minimal constructive integration process of Riemann type which includes the Lebesgue integral and also integrates the derivatives of differentiable functions. We provide a new solution to the classical problem of recovering a function from its derivative by integration, which, unlike the solution provided by Denjoy, Perron and many others, does not possess the generality which is not needed for this purpose.The descriptive version of the problem was treated by A. M. Bruckner, R. J. Fleissner and J. Foran in [2]. Their approach was based on the trivial observation that for the required minimal integral, a function F is the indefinite integral of f if and only if F'…