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

Abstract

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 $$\underset{y\to \infty }{\lim }{\int }_0^A({\int }_0^{y}f(u,v)dv)du=:I_1(A)$$ and $$\underset{x\to \infty }{\lim }{\int }_0^B({\int }_0^{x}f(u,v)du)dv=:I_2(B)$$ exist uniformly in $0<A,B<\infty$, respectively; and $$\underset{A\to \infty }{\lim }{I}_1(A)=\underset{B\to \infty }{\lim }{I}_2(B)=\underset{A,B\to \infty }{\lim }{\int }_0^A{\int }_0^{B}f(u,v)dudv.$$ 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 $f\notin L^1({\bar{\mathbb{R}}}_{+}^2)$.