Darboux Approach to Mα-integration

Abstract

It is known that one can develop Riemann integration theory via Darboux approach. The
main idea in the Darboux approach is to define an integral using upper and lower Riemann
sums. In this study we look at how -integration can be develop via Darboux approach.
Here is a brief discussion of the methodology. We define an equivalence relation on the set of
-divisions of , - such that for - divisions *(, - )+ and *(, - )+ we
say that if and only if the intervals in are exactly the intervals in . Given a
gauge on , - and a -fine division *(, - )+ of , -, we set
, - * +

Given a function on , -, and a -fine - division , we define the upper and lower
sums (respectively) in the following manner
( ) , -( ) ( )( ) and

( ) , -( ) ( )( )
provided these values exists. We were able to show that a function on , - is -
ntegrable if and only if the following exists and are equal:
(
) ∫
̅̅̅̅̅

( ) and (
) ∫

( )

In this approach we were able to prove the basic properties of the -integral. It is our next
goal to extend -integration to other spaces via Darboux approach.