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.)
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