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

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

2012-03-23

arxiv.org/abs/1203.5191v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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 $A$ 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 the following Theorem 3: If the double integral (*) converges in the regular sense, or briefly: converges regularly, then the finite limits $$\lim_{y\to \infty} \int^A_0 \Big(\int^y_0 f(u,v) dv\Big) du =: I_1 (A)$$ and $$\lim_{x\to \infty} \int^B_0 \Big(\int^x_0 f(u,v) du) dv = : I_2 (B)$$ exist uniformly in $0

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