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

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 ${\bar{\mathbb{R}}}_{+}^m:=[0,\infty {)}^m$. 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 $m,p\in {\mathbb{N}}_{+}$, then the finite limit $$\underset{v_{p+1},...,v_m\to \infty }{\lim }{\int }_{u_1}^{v_1}{\int }_{u_2}^{v_2}...{\int }_{u_p}^{v_p}{\int }_0^{v_{p+1}}...{\int }_0^{v_m}f(t_1,t_2,...,t_m)d{t}_{1}d{t}_2...d{t}_m$$ $$=:J(u_1,v_1;u_2,v_2;...;u_p,v_p),0\le u_k\le v_k<\infty ,k=1,2,...,p,$$ exists uniformly in each of its variables, and the finite limit $$\underset{v_1,v_2,...,v_p\to \infty }{\lim }J(0,v_1;0,v_2;...;0,v_p)=I$$ also exists, where $I$ is the limit of the multiple integral (**) in Pringsheim's sense.