LAPORAN PENELITIAN FUNDAMENTAL TAHUN ANGGARAN 2011
Integral Fungsi Bernilai Himpunan Pada Ruang Banach non-separable
Drs. Mohamad Muslikh, M.Si Dr. Abdul Rouf Alghofari, M.Sc
Dibiayai Oleh Direktorat Jendral Pendidikan Tinggi, Kementrian Pendidikan Nasional Melalui DIPA Universitas Brawijaya Rev. 1 Nomor: 0636/023-04.2.16/15/2011 R, tanggal 30 Maret 2011, dan Berdasarkan Surat dari DP2M DIKTI No: 121/D3/PL/2011 Tanggal 7 Pebruari 2011
UNIVERSITAS BRAWIJAYA NOVEMBER 2011
RINGKASAN Laporan hasil peneltian ini merupakan laporan hasil tahun pertama dari 2 (dua) tahun usulan penelitian yang diajukan. Oleh karena itu laporan ini hanya meliputi pengintegralan Pettis fungsi bernilai himpunan : (Ω, Σ, ) ⟶ () dengan X ruang Banach non-separable. Hasil laporan ini tidak mencakup eksistensi selektor terukur dari fungsi bernilai himpunan yang nilai-nilainya berada di dalam ruang Banach non-separable sebagaimana tujuan penelitian yang diusulkan selama dua tahun. Hasil penelitian ini merupakan perluasan dari hasil yang telah diteliti oleh para peneliti lain tentang pengintegralan Pettis untuk fungsi bernilai himpunan dalam konteks ruang Banach yang separable [ Teorema 5.1, Amri, Hess, 2000, Ziat 1997, 2000, Casttaing, Valadeir, 1977]. Upaya perluasan dari ruang Banach yang separable menjadi non-separable, diperlukan syarat lain pada ruang Banach-nya, sehingga pengintegralan Pettis fungsi bernilai himpunan dapat dikerjakan. Berdasarkan hasil penelitian yang telah dilakukan, diperoleh bahwa ruang Banach X harus mempunyai sifat SMSP- ( scalarly measurable selector property-), yaitu untuk setiap fungsi bernilai himpunan : (Ω, Σ, ) ⟶ () mempunyai selektor-selektor yang terukur secara skalar. Disamping itu ruang Banach X juga harus mempunyai sifat PIP- ( Pettis integral property) , yaitu setiap selektor-selektor di dalam F yang terukur secara skalar dan terbatas secara skalar terintegral Pettis terhadap ukuran . Dalam penelitian ini ditinjau fungsi bernilai himpunna : (Ω, Σ, ) ⟶ () dimana (Ω, Σ, ) merupakan ruang ukuran berhingga yang lengkap dan () adalah koleksi semua himpunan bagian tak kosong yang kompak-konveks-weakly dari ruang Banach X. Untuk fungsi bernilai himpunan : (Ω, Σ, ) ⟶ () dan ∈ Σ, dibangun koleksi (, ) = ∣ !" #! " $ # % # &.
Hasil penelitian menunjukkan bahwa untuk setiap fungsi bernilai himpunan : (Ω, Σ, ) ⟶ () yang terintegral Pettis dengan ruang Banach X non-separable, selalu memuat selektorselektor yang terintegral Pettis dari F ( Teorema 5.3). Berikutnya memperlihatkan bahwa untuk setiap himpunan ∈ Σ, integral Pettis , sepadan (coincide) dengan penutup (closure) dari (, ) (Teorema 5.4). Pada hasil berikutnya, diperkenalkan koleksi '( = )* ∗ (, ∗ , ) ∣ , ∗ ∈ -. ∗ / ⊂ ℝ2 , ∗ (, ∗ dimana fungsi * , ) ∶ Ω ⟶ ℝ didefinisikan sebagai * ∗ (, ∗ , )(4) = * ∗ (, ∗ , (4)) untuk setiap 4 ∈ Ω dan , ∗ ∈ ∗ . Untuk ruang Banach X separable telah diperoleh bahwa F terintegral Pettis jika dan hanya jika koleksi '( = )* ∗ (, ∗ , ) ∣ , ∗ ∈ -. ∗ / terintegral seragam. Dengan mengganti X non-separable, hasil penelitian memperlihatkan bahwa ekivalensi tersebut tetap berlaku asalkan ruang Banach X mempunyai sifat SMSP- ( scalarly measurable selector property-) dan sifat PIP- ( Pettis integral property- ). Hal ini ditunjukkan olehTeorema 5. 5 dan Akibat 5.7.
Summary Report the results of this research is the first year report of 2 (two) years of research proposals submitted. Therefore, this report covers only the set-valued function Pettis integration with X nonseparable Banach spaces. The results of this report does not include the existence of measurable selectors of the set-valued function whose values are in non-separable Banach spaces as the proposed research goals for two years. The results of of this research is an extension of the results that have been studied by other researchers on Pettis integration to the set-valued function in the context of a separable Banach space [ Theorem 5.1, Amri, Hess, 2000, Ziat 1997, 2000, Casttaing, Valadeir, 1977 ]. Extension of a separable Banach space into non-separable, the other conditions required add on its Banach space, so that Pettis integration of the set-valued function can still be done. Based on the results of research that has been conducted, found that a Banach space X has the Scalarly Measurable Selector Property with respect to , shortly − 7$, that is for every scalarly measurable set-valued function : (Ω, Σ, ) ⟶ () has a scalarly measurable selector. Besides Banach space X also must have the − $ # #! " $ % (shortly − $8$, that is every scalarly measurable and scalarly bounded selector is Pettis integrable. In this research studied the set-valued function : (Ω, Σ, ) ⟶ (), which (Ω, Σ, ) is a complete finite measure space and () is a collection all nonempty subset of a compact-convexweakly of Banach space X. To a set valued function : (Ω, Σ, ) ⟶ () and ∈ Σ, constructed collection (, ) = ∣ !" #! " $ # % # &.
The results showed that for every set-valued function : (Ω, Σ, ) ⟶ () is Pettis integrable on Banach space X with non-separable, always contain selectors of F Pettis integrable (Theorem 5.3). Next show that for any set ∈ Σ. The Pettis integral coincide with the closure of (, ) (Theorem 5.4). In the next result, introduced a collection where the function * ∗ (, ∗ , ) ∶ Ω ⟶ ℝ is defined as
'( = )* ∗ (, ∗ , ) ∣ , ∗ ∈ -. ∗ / ⊂ ℝ2 ,
* ∗ (, ∗ , )(4) = * ∗ (, ∗ , (4)) for each 4 ∈ Ω and , ∗ ∈ ∗ .
For separable Banach space X has been obtained that the Pettis integral F if and only if the collection '( = )* ∗ (, ∗ , ) ∣ , ∗ ∈ -. ∗ / uniformly integrable. By replacing X non-separable, the results of of this research show that the equivalence remains valid as long as the Banach space X has the µ-SMSP properties ( Scalarly Measurable Selector Property ) and the properties of PIP-µ ( Pettis Integral Property-µ ). This is shown in Theorem 5. 5 and 5.7.
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