Teorema Fubini

Dalam cabang analisis matematika, teorema Fubini, yakni sebuah teorema yang diperkenalkan oleh Guido Fubini, adalah sebuah teorema yang memberikan syarat kapan mungkin untuk menghitung integral ganda dengan menggunakan integral teriterasi. Pada intinya, urutan pengintegrasian boleh diganti urutannya dengan syarat integral ganda yang tersebut menghasilkan angka yang terhingga pada saat integrannya digantikan dengan nilai mutlak integran tadi.

\int _{X}\left(\int _{Y}f(x,y)\,{\text{d}}y\right)\,{\text{d}}x=\int _{Y}\left(\int _{X}f(x,y)\,{\text{d}}x\right)\,{\text{d}}y=\int _{X\times Y}f(x,y)\,{\text{d}}(x,y)

Akibatnya, teorema ini memungkinkan untuk melakukan penggantian urutan pengintegralan pada integrasi yang berlipat. Teorema Fubini berimplikasi bahwa integral lipat dua dari sebuah fungsi dengan dua variabel adalah sama jika saja fungsi tersebut dapat diintegralkan (integrable). Teorema lain yang terkait adalah teorema Tonelli yang diperkenalkan oleh Leonida Tonnelli pada tahun 1909, hanya saja teorema ini hanya bisa digunakan untuk fungsi-fungsi yang tidak negatif, bukan pada fungsi-fungsi yang dapat diintegralkan seperti pada teorema Fubini.^[1]

Sejarah

Sebelumnya, kasus khusus dari teorema Fubini untuk fungsi yang kontinu pada perkalian dari subset yang terbatas dan tertutup dari sebuah ruang vektor riil sudah diketahui oleh Euler pada abad ke 18. Pada tahun 1904, Lebesgue memperluas penemuan Euler ini kepada fungsi yang terukur dan terbatas pada perkalian interval.^[2] Kemudian, pada tahun 1906 Levi menduga, bahwa teorema temuan Lebesgue ini dapat diperluas lebih lanjut pada fungsi-fungsi yang bisa diintegralkan, bukan hanya fungsi-fungsi yang terbatas. Inilah yang dibuktikan oleh Fubini pada tahun 1907.^[3] Tonelli pada tahun 1909 memberikan sebuah variasi lain dari teorema Fubini yang diaplikasikan pada fungsi tak-negatif.^[4]

Referensi

^ Tonelli, Leonida (1909). "Sull'integrazione per parti". Atti della Accademia Nazionale dei Lincei. (5). 18 (2): 246–253.
^ Lebesgue, Henri (1904), Leçons sur l'intégration et la recherche des fonctions primitives, Paris: Gauthier-Villars
^ Fubini, Guido (1907), "Sugli integrali multipli", Rom. Acc. L. Rend. (5), 16 (1): 608–614, JFM 38.0343.02 Reprinted in Fubini, G. (1958), Opere scelte, 2, Cremonese, hlm. 243–249
^ Tonelli, Leonida (1909). "Sull'integrazione per parti". Atti della Accademia Nazionale dei Lincei. (5). 18 (2): 246–253.

Bacaan lebih lanjut

DiBenedetto, Emmanuele (2002), Real Analysis, Birkhäuser Advanced Texts: Basler Lehrbücher, Boston: Birkhäuser, doi:10.1007/978-1-4612-0117-5, ISBN 0-8176-4231-5, MR 1897317
Billingsley, Patrick (1995), "Product Measure and Fubini's Theorem", Probability and Measure, New York: Wiley, hlm. 231–240, ISBN 0-471-00710-2
Weir, Alan J. (1973), "Fubini's Theorem", Lebesgue Integration and Measure, Cambridge: Cambridge University Press, hlm. 83–92, ISBN 0-521-08728-7

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