Order-type Henstock and McShane integrals in banach lattice setting

Henstock-type integrals are studied for functions defined in a compact metric space T endowed with a regular σ-addltive measure μ, and taking values in a Banach lattice X. In particular, the space [0,1] with the usual Lebesgue measure is considered. The norm- and the order-type integral are compared and interesting results are obtained when X is an L-space. 2010 AMS Mathematics Subject Classification: 28B20, 46G10.

[1]  Douglas S. Kurtz.  Theories of Integration , 2004 .

[2]  Shizuo Kakutani,et al.  Concrete Representation of Abstract (L)-Spaces and the Mean Ergodic Theorem , 1941 .

[3]  Vladimir Kadets,et al.  The Pettis integral for multi-valued functions via single-valued ones☆ , 2007 .

[4]  A. R. Sambucini,et al.  Some Inequalities in Classical Spaces with Mixed Norms , 2002 .

[5]  J. Rodríguez.  SOME EXAMPLES IN VECTOR INTEGRATION , 2009, Bulletin of the Australian Mathematical Society.

[6]  C. Labuschagne,et al.  Discrete stochastic integration in Riesz spaces , 2010 .

[7]  A. R. Sambucini,et al.  A McShane integral for multifunctions , 2004 .

[8]  A. R. Sambucini,et al.  Filter Convergence and Decompositions for Vector Lattice-Valued Measures , 2014, 1401.7818.

[9]  J. Rodríguez.  On the equivalence of McShane and Pettis integrability in non-separable Banach spaces , 2008 .

[11]  A. R. Sambucini,et al.  A NOTE ON COMPARISON BETWEEN BIRKHOFF AND MCSHANE-TYPE INTEGRALS FOR MULTIFUNCTIONS , 2015, 1509.04484.

[12]  C. Swartz,et al.  Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane , 2004 .

[13]  A. Boccuto,et al.  Nikodým-Type Theorems for Lattice Group-Valued Measures with Respect to Filter Convergence , 2014 .

[14]  A. R. Sambucini,et al.  Vitali-type theorems for filter convergence related to vector lattice-valued modulars and applications to stochastic processes , 2014, 1401.7835.

[15]  YE Guo-ju.  The Henstock Integral of Banach-valued Functions , 2006 .

[16]  On the equivalence of McShane and Pettis integrability in non-separable Banach spaces , 2008 .

[17]  T. Jech.  Measure Algebras , 2017, 1705.01000.

[18]  D. Fremlin,et al.  On the integration of vector-valued functions , 1992, math/9202202.

[19]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[20]  A. Bukhvalov,et al.  BANACH LATTICES - SOME BANACH ASPECTS OF THEIR THEORY , 1979 .

[21]  D. Fremlin.  The generalized McShane integral , 1995 .

[22]  L. Di Piazza,et al.  The McShane, PU and Henstock integrals of Banach valued functions , 2002 .

[23]  A. R. Sambucini,et al.  A note on set-valued Henstock-McShane integral in Banach (lattice) space setting , 2014, 1405.6530.

[24]  D. Fremlin.  The Henstock and McShane integrals of vector-valued functions , 1994 .

[25]  J. K. Hunter,et al.  Measure Theory , 2007 .

[26]  J. Rodríguez.  On the existence of Pettis integrable functions which are not Birkhoff integrable , 2004 .

[27]  A. P. Solodov,et al.  A Variational Integral for Banach-Valued Functions , 1998 .

[28]  Integration in Hilbert generated Banach spaces , 2010 .

[29]  L. Piazza,et al.  A characterization of variationally McShane integrable Banach-space valued functions , 2001 .

[30]  D. Preiss,et al.  When do McShane and Pettis integrals coincide , 2003 .

[31]  P. C. Lee,et al.  The Integral: An Easy Approach after Kurzweil and Henstock , 2001 .