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

TL;DR

This paper extends Fubini's theorem to certain functions that are locally integrable but not globally integrable, establishing conditions under which double integrals converge and can be computed as iterated integrals.

Contribution

The paper proves a generalized Fubini's theorem for functions in local L^1 that are not globally integrable, clarifying convergence conditions for double integrals.

Findings

Regular convergence implies uniform limits of iterated integrals.

Double integral convergence ensures the equality of iterated limits.

The result applies to functions outside the classical L^1 space.

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 $$\lim_{y\to \infty} \int^A_0 \Big(\int^y_0 f(u,v) dv\Big) du =: I_1 (A)$$ and $$\lim_{x\to \infty} \int^B_0 \Big(\int^x_0 f(u,v) du) dv = : I_2 (B)$$ exist uniformly in $0<A, B <\infty$, respectively; and $$\lim_{A\to \infty} I_1(A) = \lim_{B\to \infty} I_2 (B) = \lim_{A, B \to \infty} \int^A_0 \int^B_0 f(u,v) du dv.$$ 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 $f\not\in L^1 (\bar{\R}^2_+)$.