From Surf Wiki (app.surf) — the open knowledge base

Pfeffer integral

Definition of mathematical integration

In mathematics, the Pfeffer integral is an integration technique created by Washek Pfeffer as an attempt to extend the Henstock–Kurzweil integral to a multidimensional domain. This was to be done in such a way that the fundamental theorem of calculus would apply analogously to the theorem in one dimension, with as few preconditions on the function under consideration as possible. The integral also permits analogues of the chain rule and other theorems of the integral calculus for higher dimensions.

Definition

The construction is based on the Henstock or gauge integral, however Pfeffer proved that the integral, at least in the one dimensional case, is less general than the Henstock integral. It relies on what Pfeffer refers to as a set of bounded variation, this is equivalent to a Caccioppoli set. The Riemann sums of the Pfeffer integral are taken over partitions made up of such sets, rather than intervals as in the Riemann or Henstock integrals. A gauge is used, exactly as in the Henstock integral, except that the gauge function may be zero on a negligible set.

Properties

Pfeffer defined a notion of generalized absolute continuity ACG^, close to but not equal to the definition of a function being ACG_, and proved that a function is Pfeffer integrable if it is the derivative of an ACG^* function. He also proved a chain rule for the Pfeffer integral. In one dimension his work as well as similarities between the Pfeffer integral and the McShane integral indicate that the integral is more general than the Lebesgue integral and yet less general than the Henstock–Kurzweil integral.

Bibliography

{{Citation
{{Citation

Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

definitions-of-mathematical-integration

Want to explore this topic further?

Ask Mako anything about Pfeffer integral — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report