IJPAM: Volume 66, No. 4 (2011)

SOME APPLICATIONS OF PRINCIPLE OF DIVISION

Gheorghe Chivu
``Dimitrie Cantemir" University of Târgu Mures
3-5, Bodoni Sándor Street, 540 500, ROMANIA
e-mail: chivu_1955@yahoo.com


Abstract.A survey of known results, from integration theory is presented, showing that many theorems can be proved using a ``principle of division''.

Let $\lambda$ denote Lebesque measure on $\IR^k$. If $X$ is a compact set in $\IR^k,$ if $F$ is a Banach space, and if $f\in C(X, F)$, then the Bochner integral of $f$ is the unique set function from $\mathcal{K}(X)$ to $F$ which is $\lambda$-additive and which fulfills an infinitesimal condition with respect to $f.$ Finally we use the ``Principle of division'' to prove a changing variables theorem.

Received: January 3, 2011

AMS Subject Classification: 31D05, 60J45

Key Words and Phrases: compact set, $\lambda$-division, basis of compact sets, additive set functions, subadditive set functions, basis of filter, infinitesimal condition, principle of definition

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2011
Volume: 66
Issue: 4