TY - JOUR
AU - Racca, Abraham P.
PY - 2019/12/18
Y2 - 2021/06/20
TI - Darboux Approach to Mα-integration
JF - Abstract Proceedings International Scholars Conference
JA - isc
VL - 7
IS - 1
SE -
DO - 10.35974/isc.v7i1.1069
UR - https://jurnal.unai.edu/index.php/isc/article/view/1069
SP - 1871-1878
AB - <p>It is known that one can develop Riemann integration theory via Darboux approach. The<br>main idea in the Darboux approach is to define an integral using upper and lower Riemann<br>sums. In this study we look at how -integration can be develop via Darboux approach.<br>Here is a brief discussion of the methodology. We define an equivalence relation on the set of<br>-divisions of , - such that for - divisions *(, - )+ and *(, - )+ we<br>say that if and only if the intervals in are exactly the intervals in . Given a<br>gauge on , - and a -fine division *(, - )+ of , -, we set<br>, - * +</p><p>Given a function on , -, and a -fine - division , we define the upper and lower<br>sums (respectively) in the following manner<br>( ) , -( ) ( )( ) and</p><p>( ) , -( ) ( )( )<br>provided these values exists. We were able to show that a function on , - is -<br>ntegrable if and only if the following exists and are equal:<br>(<br>) ∫<br>̅̅̅̅̅</p><p>( ) and (<br>) ∫</p><p>( )</p><p>In this approach we were able to prove the basic properties of the -integral. It is our next<br>goal to extend -integration to other spaces via Darboux approach.</p>
ER -