On the convergence of double integrals and a generalized version of Fubini’s theorem on successive integration
Tóm tắt
Let the function
$$f: \overline {\mathbb{R}}_+^2 \rightarrow \mathbb{C}$$
be such that
$$f \in L_{loc}^1(\overline{\mathbb{R}}_+^2)$$
. We investigate the convergence behavior of the double integral
$$(*) \;\;\;\;\;\;\;\;\;\;\; \int_{0}^{A} \int_{0}^{B} f(u, v)dudv \;\;\; as \;\; A,B \rightarrow \infty$$
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
$$\lim_{y \rightarrow \infty} \int_{0}^{A} \left(\int_{0}^{y} f(u, v)dv \right) du =: I_1(A) \;and \lim_{x \rightarrow \infty} \int_{0}^{B} \left( \int_{0}^{x} f(u, v)du \right) dv =: I_2(B)$$
exist uniformly in 0 < A, B < ∞, respectively, and
$$\lim_{A \rightarrow \infty} I_1(A) = \lim {_{B\rightarrow \infty}} I_2(B) = \lim_{A,B \rightarrow \infty} \int_{0}^{A} \int_{0}^{B} f(u, v)dudv.$$
This can be considered as a generalized version of Fubini’s theorem on successive integration when
$$f \in L_{loc}^1(\overline{\mathbb{R}}_+^2)$$
, but
$$f \notin L^1(\overline{\mathbb{R}}_+^2)$$
.
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