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

R I N G K A S A N

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 selektor-

selektor 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.

S u m m a r y

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 non-

separable 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-convex-

weakly 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 '( = )*∗(,∗, �) ∣ ,∗ ∈ -.∗/ ⊂ ℝ2, where the function *∗(,∗, �) ∶ Ω ⟶ ℝ is defined as

*∗(,∗, �)(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.

DAFTAR PUSTAKA

R.J. Aumann, 1965, Integrals of set valued function, Journal of Mathematical Analysis and application 12,

1-12.

Z. Artstein and J.A Burns. 1975, Integrations of compact set-valued functions, Pacific Journal of

Mathematics vol 58, No. 2, 297-307.

C. Castaing, M.Valadier, 1977, Convex analysis and measurable multifunctions, Lecture notes in

Math.,vol.580,Springer-Verlag, Berlin,MR 57 #7169.

B. Cascales, V. Kadets, J. Rodriguez, 2004, Brikhoff integral for multi-valued functions, J. Math. Anal.

Appl. 297, no. 2, 540-560.

A. Coste, 1974, Set-Valued measures, Proceeding of the Conference Topology and Measure, Part I

(Greifswald) , 55-74.

G. Debrue,1967,Integration of correspondences, in Proceedings of the Fifth Berkeley Symposium on

Mathematical Statistic and Probability 2, 351-372

L Di Piazza, 2005, Set-Valued Kurzweil-Henstock-Pettis integral, Set-Valued Analysis 13, 167-179.

K. El Amri, C. Hess, 2000, On the Pettis integral of closed valued multifunctions, Set-Valued Anal. 8, 329-

360, MR 2002e:26025

G.A. Edgar, 1977, Measurability in Banach space, Indiana Univ. Math. J, 26, no 4, 663-677.

D.H. Fremlin, M. Talagrand, 1979, A Decomposition Theorem for Additive set-function with applications

to Pettis Integrals and argodic means , Math Z, 168, no. 2, 117-142.

K. Floret, 1980, Weakly compact set, Lecture notes in Mathematics, vol 801, Springer, Berlin.

C. Hess, 2002, Set-valued integration and set-valued probability theory, Handbook of Measure Theory,

vols I,II, North-Holland, Amsterdam.,617-673. MR 1954624

C. Hess, H. Ziat, 2002, Theoreme de Komlos pour des multifonctions integrables au sens de Pettis et

applications, Ann. Sci. Math. Quebec,26, 181-198, MR 1 980 843.

E. Klein and Thompson,1984, Theory of Correspondences, texts, John Wiley and Sons,New York.,MR

86a:90012

K. Kuratowski, C, Ryll-Nardzewski, 1965, A General Theorem on selectors, Bull. Acad. Polon. Sci. Ser. Sci.

Math. Asrtonom. Phys. 13.

M. Muslikh, 2007, On the Birkhoff integral of closed bounded valued multifunction, Proc, SEAMS,

Jogyakarta.

M. Valadier, 1971, Multi-Applications measurable a valeurs convexes compactes, J. Math. Pures. Appl.(9)

50, 265-297.

V. Zizler, 2003, Nonseparable Banach spaces, Handbook of The Geometry of Banach spaces, Vol. 2,

North-Holland, Amsterdam, pp 1743-1816.

H. Ziat, 1997, Convergence theorems for Petis integrable multifunctions, Bull. Polish Acad. Sci. Math. 45,

123-137, MR98i:46035.

H. Ziat, 2000, On a characterization of Pettis integrable multifunctions, Bull. Polish

Acad. Math. 48, 227-230, MR 2001d:46063