On some properties of the space of tensor integrable functions (Q925140)

The authors study the concept of integrating \(X\)-valued \(\nu\)-measurable functions which is due to \textit{G.\,F.\thinspace Stefánsson} [Ill.\ J.\ Math.\ 45, No.\,3, 925--938 (2001; Zbl 0993.46024)]. Here, \(X,Y\) are Banach spaces and \(\nu\) is a \(Y\)-valued countably additive vector measure. The integral is defined in a similar way as this is done for the Bochner integral approximating the function by simple functions and completing the space of simple functions via a special seminorm. The space \(L_1(\nu,X,Y)\) is introduced and various of its properties are presented.