abstract:In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions. The dominated convergence theorem does not hold for the Riemann integral because the limit of a sequence of Riemann-integrable functions is in many cases not Riemann-integrable.
In the present note we propose to deducethisresultdirectly from thedominatedconvergence theoremofstochasticintegrals, so our proof is much simpler than the original one.