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 yββlimββ«0Aβ(β«0yβf(u,v)dv)du=:I1β(A) and xββlimββ«0Bβ(β«0xβf(u,v)du)dv=:I2β(B) exist uniformly in 0<A,B<β, respectively;
and AββlimβI1β(A)=BββlimβI2β(B)=A,Bββlimββ«0Aββ«0Bβf(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 fξ βL1(RΛ+2β)