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The McShane, PU and Henstock integrals of Banach valued functions

Luisa Di Piazza, Valeria Marraffa (2002)

Czechoslovak Mathematical Journal

Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational integrals...

The monotone convergence theorem for multidimensional abstract Kurzweil vector integrals

Márcia Federson (2002)

Czechoslovak Mathematical Journal

We prove two versions of the Monotone Convergence Theorem for the vector integral of Kurzweil, ∫ R d α ( t ) f ( t ) , where R is a compact interval of ℝ n , α and f are functions with values on L ( Z , W ) and Z respectively, and Z and W are monotone ordered normed spaces. Analogous results can be obtained for the Kurzweil vector integral, ∫ R α ( t ) d f ( t ) , as well as to unbounded intervals R .

The s-Perron, sap-Perron and ap-McShane integrals

Joo Bong Kim, Deok Ho Lee, Woo Youl Lee, Chun-Gil Park, Jae Myung Park (2004)

Czechoslovak Mathematical Journal

In this paper, we study the s-Perron, sap-Perron and ap-McShane integrals. In particular, we show that the s-Perron integral is equivalent to the McShane integral and that the sap-Perron integral is equivalent to the ap-McShane integral.

The symmetric Choquet integral with respect to Riesz-space-valued capacities

Antonio Boccuto, Beloslav Riečan (2008)

Czechoslovak Mathematical Journal

A definition of “Šipoš integral” is given, similarly to [3],[5],[10], for real-valued functions and with respect to Dedekind complete Riesz-space-valued “capacities”. A comparison of Choquet and Šipoš-type integrals is given, and some fundamental properties and some convergence theorems for the Šipoš integral are proved.

The Vitali convergence theorem for the vector-valued McShane integral

Richard Reynolds, Charles W. Swartz (2004)

Mathematica Bohemica

The classical Vitali convergence theorem gives necessary and sufficient conditions for norm convergence in the space of Lebesgue integrable functions. Although there are versions of the Vitali convergence theorem for the vector valued McShane and Pettis integrals given by Fremlin and Mendoza, these results do not involve norm convergence in the respective spaces. There is a version of the Vitali convergence theorem for scalar valued functions defined on compact intervals in ℝ n given by Kurzweil and...

The weak McShane integral

Mohammed Saadoune, Redouane Sayyad (2014)

Czechoslovak Mathematical Journal

We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a σ -finite outer regular quasi Radon measure space ( S , Σ , 𝒯 , μ ) into a Banach space X and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function f from S into X is weakly McShane integrable on each measurable subset of S if and only if...

Currently displaying 21 – 40 of 46