On the regular convergence of multiple series of numbers and multiple integrals of locally integrable functions over ℝ̄n +n n m

摘要

We investigate the regular convergence of the m-multiple series $$sumlimits_{j_1 = 0}^infty {sumlimits_{j_2 = 0}^infty cdots sumlimits_{j_m = 0}^infty {c_{j_1 ,j_2 } , ldots j_m } }$$ (*) of complex numbers, where m ≥ 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 also be computed by successive summation. We introduce and investigate the regular convergence of the m-multiple integral $$int_0^infty {int_0^infty { cdots int_0^infty {fleft( {t_1 ,t_2 , ldots ,t_m } right)dt_1 } } } dt_2 cdots dt_m ,$$ (**) where f : ℝ̄ + m → ℂ is a locally integrable function in Lebesgue’s sense over the closed nonnegative octant ℝ̄ + m := [0,∞) m . Our main result is a generalized version of Fubini’s theorem on successive integration formulated in Theorem 4.1 as follows. If f ∈ L loc 1 (ℝ̄ + m ), the multiple integral (**) converges regularly, and m = p + q, where p and q are positive integers, then the finite limit $$mathop {lim }limits_{v_{_{p + 1} } , cdots ,v_m to infty } int_{u_1 }^{v_1 } {int_{u_2 }^{v_2 } { cdots int_0^{v_{p + 1} } { cdots int_0^{v_m } {fleft( {t_1 ,t_2 , ldots t_m } right)dt_1 dt_2 } cdots dt_m = :Jleft( {u_1 ,v_1 ;u_2 v_2 ; ldots ;u_p ,v_p } right)} , 0 leqslant u_k leqslant v_k < infty } ,k = 1,2, ldots p,}$$ exists uniformly in each of its variables, and the finite limit $$mathop {lim }limits_{v_1 ,v_2 cdots ,v_p to infty } Jleft( {0,v_1 ;0,v_2 ; ldots ;0,v_p } right) = I$$ also exists, where I is the limit of the multiple integral (**) in Pringsheim’s sense. The main results of this paper were announced without proofs in the Comptes Rendus Sci. Paris (see [8] in the References).