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On the convergence of double integrals and a generalized version of Fubini's theorem on successive integration
Let the function f: \bar{\R}^2_+ \to \C be such that f\in L^1_{\loc}
(\bar{\R}^2_+). We investigate the convergence behavior of the double integral
\int^A_0 \int^B_0 f(u,v) du dv \quad {\rm as} \quad A,B \to
\infty,\leqno(*) where and 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 the following Theorem 3: If the
double integral (*) converges in the regular sense, or briefly: converges
regularly, then the finite limits and exist uniformly in , respectively;
and This can be considered as a
generalized version of Fubini's theorem on successive integration when f\in
L^1_{\loc} (\bar{\R}^2_+), but
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