Left-endpoint approximation of a continuous semimartingale integral (${\sum }_{k<\lfloor t/h\rfloor }{H}_{kh}(X_{(k+1)h}-X_{kh})\stackrel{u.c.p.}{\to }{\int }_0^t{H}_{s}d{X}_s$)

For a left-continuous locally bounded adapted process $H$ and a continuous semimartingale $X$, the elementary predictable left-endpoint approximations tend pointwise to $H$. Decompose $X$ into a continuous local martingale and a continuous finite-variation process. After localization, dominated convergence with respect to the variation measure handles the finite-variation integral. The Itô isometry and Doob L2 maximal inequality handle the martingale integral. The omitted partial last interval is bounded by the local bound on $H$ times the modulus of continuity of $X$. This proves uniform convergence on compacts in probability and is the grid-sum foundation of the Itô product rule.