We consider the Riemann sum of the same type we introduced at the MSJ autumn meeting in 2020 and discuss approximation and characterization of the stochastic integral with respect to a local martingale by the Riemann sum. We extend the preceding result given at the meeting mentioned above mainly in the following respect: (1) on the weight measures defining the Riemann sum, (2) on the continuity of the integrator and (3) on the uniform Riemann approximation of the stochastic integral.