The limit problem of function sequence in generalized integral is studied. With the limit theory of function sequence，the limit theorem of Riemann integral function sequence is given within the uniform bounded and inner-closed uniformly convergent function sequence.Undering the conditions of uniform convergence of generalized integral of the function sequence，the limit theorem and control convergent theorem of function sequence integral under the generalized integral are obtained.

%U https://zkxb.jsu.edu.cn/EN/10.3969/j.cnki.jdxb.2016.06.001