Convergence and Term-by-Term Integrability of Function Sequences

Let us consider the integral \(I_n = \int ^{l}_{0}f(\alpha )\varphi (\alpha ,n)\,\text {d}\alpha \) , formed with a given kernel \(\varphi (\alpha ,n)\) and a function \(f(\alpha )\) . We want to find its limit as n approaches infinity.