Pfeffer integral
Encyclopedia
In mathematics, the Pfeffer integral is an integration technique created by Washek Pfeffer
as an attempt to extend the Henstock 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.
. 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.
, close to but not equal to the definition of a function being 
, and proved that a function is Pfeffer integrable iff it is the derivative of an 
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 integral.
Washek Pfeffer
Washek F. Pfeffer is a Czech-born US mathematician and Emeritus Professor at the University of California, Davis. Pfeffer is one of the world's pre-eminent authorities on real integration and has authored several books on the topic of integration, and numerous papers on these topics and others...
as an attempt to extend the Henstock integral to a multidimensional domain. This was to be done in such a way that the fundamental theorem of calculus
Fundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
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 setCaccioppoli set
In mathematics, a Caccioppoli set is a set whose boundary is measurable and has a finite measure. A synonym is set of finite perimeter...
. 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, close to but not equal to the definition of a function being , and proved that a function is Pfeffer integrable iff it is the derivative of an 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 integral.