This paper presents several results on the Fatou property of quasiconvex law-invariant functionals defined on a rearrangement invariant space X. First , we show that for any proper quasiconvex law-invariant functional \rho on \mathcal{X \sigma(X,X_n^\sim) -lower semicontinuity and \sigma(X,L^\infty) -lower semicontinuity of \rho are equivalent, where \mathcal{X}_n^\sim is the order continuous dual of \mathcal{X}. Second, we provide some relations between the Fatou property and some other types of lower semicontinuity, namely norm lower semicontinuity and the strong Fatou property. In particular, we show that when \mathcal{X\neq L^1, the strong Fatou property and the Fatou property coincide. Finally, we generalize some extension results by Gao et al. }%DIFDELCMD < [%%% 9 to a general rearrangement invariant space .
This paper presents relations between several types of closedness of a law-invariant convex set in a rearrangement invariant space X. In particular , we show that order closedness, \sigma(X,X_n^\sim) -closedness and \sigma(X,L^\infty) -closedness of a law-invariant convex set in \mathcal{X are equivalent, where \mathcal{X}_n^\sim is the order continuous dual of \mathcal{X}. We also provide some application to proper quasiconvex law-invariant functionals with the Fatou property \neq L^1, the strong Fatou property and the Fatou property coincide. Finally, we generalize some extension results by Gao et al. }%DIFDELCMD < [%%% .
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several results on the Fatou property of quasiconvex
relations between several types of closedness of a
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functionals defined on
convex set in
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First
In particular
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for any proper quasiconvex law-invariant functional \rho on \mathcal{X
order closedness,
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-lower semicontinuity
-closedness
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-lower semicontinuity of \rho
-closedness of a law-invariant convex set in \mathcal{X
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Second, we provide some relations between
We also provide some application to proper quasiconvex law-invariant functionals with
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and some other types of lower semicontinuity, namely norm lower semicontinuity and the strong Fatou property. In particular, we show that when \mathcal{X
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9
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to a general rearrangement invariant space