========================= Remarks: - Cesare Arzela published his theorem in 1885 (see [2]) by considering that (f_n), is a sequence of Riemann integrable functions. - It remained almost unnoticed until it was rediscovered independently in 1897 by W.F. Osgood [9], who stated it, however,only for continuous functions. In this special case it is customary to call it Osgood's Theorem. Therefore see ...[9], for OSGOOD THEOREM. For other questions regarding these two theorems (Arzela , Osgood), see the works listed below. REFERENCES : [1] ALEXANDROV P.S., ,, On quasi-uniform convergence" (Russian), Uspehi Mat.Nauk vol.1(23),(1948),213-215. [2] ARZELA C., ,,Sulla integrazione per serie", Rendinconti Accad.Lincei Roma , 1 (1885), 532-537 , 566-569 [3] ARZELA C., ,,Sulle serie di funzioni", (parte seconda), Memorie Accad.Sci. Bologna 8(1900) 701-744. (see pp.723-724). [4] BOREL E., ,,Lecons sur les fonctions de variables reeles", Paris , Gauthier-Villars,11905,(see p.41). [5] GAGAEFF B.,,, Sur les suites convergentes de fonctions mesurables B ", Fundamenta Mathematicae , vol. XVIII,(1932) , 182-188. [6] HOBSON E.W., ,, The theory of functions of a real variabel and the Theory of Fourier's series" , t.II, 2-end ed., 1926, 131-132. [7] LEBESGUE H., Sur l'integration des fonctions discontinues, Annales Ecole Norm.Sup.,(3) 1 , 1910, 361-450.(seee page 375) [8] LEVI Beppo , ,, Sopra l'integrazione delle serie ", Rend.Instituto Lombardo di Sci. e Lett., (2) 39 (1906), 775-780. [9] OSGOOD W.F., ,,Non-uniform convergence and the integration of series term by term ", Amer.J.Math., 19 (1897) ,155-190.
