### A Cauchy type condition and integrability

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An exact Radon-Nikodym derivative is obtained for a pair (I,J) of positive linear functionals, with J absolutely continuous with respect to I, using a notion of exhaustion of I on elements of a function algebra lattice.

We prove that the spectral sets of any positive abstract Riemann integrable function are measurable but (at most) a countable amount of them. In addition, the integral of such a function can be computed as an improper classical Riemann integral of the measures of its spectral sets under some weak continuity conditions which in fact characterize the integral representation.

We present a new characterization of Lebesgue measurable functions; namely, a function f:[0,1]→ ℝ is measurable if and only if it is first-return recoverable almost everywhere. This result is established by demonstrating a connection between almost everywhere first-return recovery and a first-return process for yielding the integral of a measurable function.

It is commonly known that absolute gauge integrability, or Henstock-Kurzweil (H-K) integrability implies Lebesgue integrability. In this article, we are going to present another proof of that fact which utilizes the basic definitions and properties of the Lebesgue and H-K integrals.

A new definition of the product integral is given. The definition is based on a procedure which is analogous to the sum definition of the Bochner integral given by J. Kurzweil and E.J. McShane. The new definition is shown to be equivalent to the seemingly verey different one given by J.D. Dollard and C.N. Friedman in [1] and [2].