On the regular convergence of multiple series of numbers and multiple integrals of locally integrable functions over $\bar{\R}^m_+$

TL;DR

This paper establishes conditions under which multiple series and integrals of locally integrable functions converge regularly, allowing for successive summation and integration, thereby generalizing Fubini's theorem to these settings.

Contribution

It proves a generalized Fubini's theorem for regular convergence of multiple series and integrals of locally integrable functions.

Findings

Regular convergence allows successive summation of multiple series.

Multiple integrals of locally integrable functions converge regularly under certain conditions.

The paper extends Fubini's theorem to the setting of regular convergence for multiple integrals.

Abstract

We investigate the regular convergence of the $m$-multiple series $$\sum^\infty_{j_1=0} \sum^\infty_{j_2=0}...\sum^\infty_{j_m=0} \ c_{j_1, j_2,..., j_m}\leqno(*)$$ of complex numbers, where $m\ge 2$ is a fixed integer. We prove Fubini's theorem in the discrete setting as follows. If the multiple series (*) converges regularly, then its sum in Pringsheim's sense can be computed by successive summation. We introduce and investigate the regular convergence of the $m$-multiple integral $$\int^\infty_0 \int^\infty_0...\int^\infty_0 f(t_1, t_2,..., t_m) dt_1 dt_2...dt_m,\leqno(**)$$ where $f: \bar{\R}^m_+ \to \C$ is a locally integrable function in Lebesgue's sense over the closed positive octant $\bar{\R}^m_+:= [0, \infty)^m$. Our main result is a generalized version of Fubini's theorem on successive integration formulated in Theorem 4.1 as follows. If $f\in L^1_{\loc} (\bar{\R}^m_+)$, the multiple integral (**) converges regularly, and $m=p+q$, where $m, p\in \N_+$, then the finite limit $$\lim_{v_{p+1},..., v_m \to \infty} \int^{v_1}_{u_1} \int^{v_2}_{u_2}...\int^{v_p}_{u_p} \int^{v_{p+1}}_0...\int^{v_m}_0 f(t_1, t_2,..., t_m) dt_1 dt_2...dt_m$$ $$=:J(u_1, v_1; u_2, v_2;...; u_p, v_p), \quad 0\le u_k\le v_k<\infty, \ k=1,2,..., p,$$ exists uniformly in each of its variables, and the finite limit $$\lim_{v_1, v_2,..., v_p\to \infty} J(0, v_1; 0, v_2;...; 0, v_p)=I$$ also exists, where $I$ is the limit of the multiple integral (**) in Pringsheim's sense.