MR2481817 (2010e:46040): Haluška, Ján; Hutník, Ondrej On vector integral inequalities. Mediterr. J. Math. 6 (2009), no. 1, 105–124. (Reviewer: Luisa Di Piazza),

I. Dobrakov in his papers [Czechoslovak Math. J. 40(115) (1990), no. 1, 8--24; MR1032359 (90k:46097); Czechoslovak Math. J. 40(115) (1990), no. 3, 424--440; MR1065022 (91g:46052)] developed a theory for integrating vector-valued functions with respect to operator-valued measures: Let X and Y be two Banach spaces, Δ be a δ-ring of subsets of a nonempty set T, L(X,Y) be the space of all continuous operators L:X→Y, and m:Δ→L(X,Y) be an operator-valued measure σ-additive in the strong operator topology of L(X,Y). A measurable function f:T→X is said to be integrable in the sense of Dobrakov if there exists a sequence of simple functions fn:T→X, n∈N, converging m-a.e. to f and the integrals ∫.fndm are uniformly σ-additive measures on σ(Δ) (i.e. the σ-algebra generated by Δ). The integral of the function f on E∈σ(Δ) is defined by the equality ∫Efdm=limn→∞∫Efndm. The first author, in the papers [Math. Slovaca 43 (1993), no. 2, 185--192; MR1274601 (95f:46069); Rev. Roumaine Math. Pures Appl. 38 (1993), no. 4, 327--337; MR1258045 (95g:28008); Czechoslovak Math. J. 47(122) (1997), no. 2, 205--219; MR1452416 (98h:46044)], has generalized the Dobrakov integration to the complete bornological locally convex topological vector spaces (C.B.L.C.S., for short), developing a new technique for C.B.L.C.S. and operator-valued measures. The main novelty of such a theory is that instead of the "classical'' objects such as a submeasure, a norm, a metric, etc., he needs to work with lattices of submeasures, norms, etc. In the present paper the authors first give a brief development of this new integration theory in C.B.L.C.S. (see Section 2). In Section 3 some inequalities, which are important tools in this Dobrakov-type integration technique, are proved. Reviewed by Luisa Di Piazza