ACTA issues

On the convergence of double integrals and a generalized version of Fubini's theorem on successive integration

Ferenc Móricz

Acta Sci. Math. (Szeged) 78:3-4(2012), 469-487
23/2012

 Abstract. Let the function $f\colon\overline {\msbm R}^2_+ \to{\msbm C}$ be such that $f\in L^1_{\rm loc} (\overline{\msbm R}^2_+)$. We investigate the convergence behavior of the double integral $(*)$\hskip25pt$\int ^{A\strut }_{0\strut }\int ^B_0 f(u,v) du dv {\rm as} A,B \to\infty$, where $A$ and $B$ tend to infinity independently of one another, while using two notions of convergence: that in Pringsheim's sense and that in the regular sense. Our main result is that if the double integral ($*$) converges in the regular sense, then the finite limits $\lim_{y\to\infty } \int ^{A\strut }_{0\strut }\left(\int ^y_0 f(u,v) dv\right ) du =: I_1 (A)$ and $\lim_{x\to\infty } \int ^B_0\left(\int ^x_0 f(u,v) du\right ) dv =: I_2 (B)$ exist uniformly in $0< A, B < \infty$, respectively, and \hskip15pt$\lim_{A\to\infty } I_1(A) = \lim_{B\to\infty } I_2 (B) =\lim_{A, B \to\infty } \int ^{A\strut }_{0\strut } \int ^B_0 f(u,v) du dv.$ This can be considered as a generalized version of Fubini's theorem on successive integration when $f\in L^1_{\rm loc} (\overline{\msbm R}^2_+)$, but $f\not\in L^1 (\overline{\msbm R}^2_+)$. AMS Subject Classification (1991): 28A35; 40A05, 40A10, 40B05 Keyword(s): double series of complex numbers, double integrals of locally integrable functions over $\overline{\msbm R}^2_+$ in Lebesgue's sense, convergence in Pringsheim's sense, regular convergence, absolute convergence, a generalized version of Fubini's theorem on successive integration Received April 4, 2012. (Registered under 23/2012.)