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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 multipleseries (*) converges regularly, then its sum in Pringsheim's sense can becomputed by successive summation.  We introduce and investigate the regular convergence of the $m$-multipleintegral $$\int^\infty_0 \int^\infty_0...\int^\infty_0 f(t_1, t_2,..., t_m)dt_1 dt_2...dt_m,\leqno(**)$$ where $f: \bar{\R}^m_+ \to \C$ is a locallyintegrable function in Lebesgue's sense over the closed positive octant$\bar{\R}^m_+:= [0, \infty)^m$. Our main result is a generalized version ofFubini's theorem on successive integration formulated in Theorem 4.1 asfollows. 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$$\lim_{v_{p+1},..., v_m \to \infty} \int^{v_1}_{u_1}\int^{v_2}_{u_2}...\int^{v_p}_{u_p} \int^{v_{p+1}}_0...\int^{v_m}_0 f(t_1,t_2,..., t_m) dt_1 dt_2...dt_m$$ $$=:J(u_1, v_1; u_2, v_2;...; u_p, v_p), \quad0\le u_k\le v_k<\infty, \ k=1,2,..., p,$$ exists uniformly in each of itsvariables, 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 multipleintegral (**) in Pringsheim's sense.

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