Illinois Journal of Mathematics

Integration in vector spaces

Gunnar F. Stefánsson

Abstract

We define an integral of a vector-valued function $f:\Omega\longrightarrow X$ with respect to a bounded countably additive vector-valued measure $\nu:\Sigma\longrightarrow Y$ and investigate its properties. The integral is an element of $X\check{\otimes}Y$, and when $f$ is $\nu$-measurable we show that $f$ is integrable if and only if $\|f\|\in L_{1}(\nu)$. In this case, the indefinite integral of $f$ is of bounded variation if and only if $\|f\|\in L_{1}(|\nu|)$. We also define the integral of a weakly $\nu$-measurable function and show that such a function $f$ satisfies $x^{*}f\in L_{1}(\nu)$ for all $x^{*}\in X^{*}$ and is $|y^{*}\nu|$-Pettis integrable for all $y^{*}\in Y^{*}$.

Article information

Source
Illinois J. Math., Volume 45, Number 3 (2001), 925-938.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138160

Digital Object Identifier
doi:10.1215/ijm/1258138160

Mathematical Reviews number (MathSciNet)
MR1879244

Zentralblatt MATH identifier
0993.46024

Citation

Stefánsson, Gunnar F. Integration in vector spaces. Illinois J. Math. 45 (2001), no. 3, 925--938. doi:10.1215/ijm/1258138160. https://projecteuclid.org/euclid.ijm/1258138160