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2 Bulletin of the Polish Academy of Sciences. Mathematics

2 Michalak A.

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On the Fubini theorem for the Pettis integral for bounded functions

Michalak A.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

Vol. 49, no 1

1-14

We show that if X is a WGG Banach space and it does not contain any isomorphic copy of l1, then for every bounded Pettis integrable function f : [0, 1]^2 --> X* there exists a scalarly equivalent function for which the Fubini theorem for the Pettis integral holds. On the other hand, we show that for every bounded Pettis integrable function f : [0, 1]^2 --> l^2 (R) there exists a scalarly equivalent bounded function for which the Fubini theorem for the Pettis integral does not hold. We also show (assuming the Martin axiom) that there exists a bounded Pettis integrable function f : [0, 1]^2 --> L^[infinity](lambda) such that for each function g scalarly equivalent to f the function s --> g(t, s) is not weakly measurable for almost every t [belongs to] [0, 1].

On Pettis integrability of translations of functions in L^[infinity]

Michalak A.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 2

113-119

Let K be a compact Hausdorff space, mi a positive Radon measure on K, and let G be a compact group with the Haar measure lambda. We consider properties of the following generalization of translations on groups: we associate with every bounded to mi x R[lambda]- measurable function f : K x G --> C the function T[sub f] : G --> L^[infinity] (mi x R[lambda]) given by T[sub f](t) = f[sub t] where f[sub t](r, s) = f (r, ts). We show that if K and G are metrizable and the class of f [belongs to] L^[infinity](mi x R[lambda]) contains a function g such that lambda({t [belongs to] G : (r, t) is a point of discontinuity of g}) = 0 for every r [belongs to] supp(mi), then T[sub f] is Pettis integrable with respect to lambda.