= Left-endpoint approximation of a continuous semimartingale integral
{title2=$\sum_{k<\lfloor t/h\rfloor}H_{kh}(X_{(k+1)h}-X_{kh})\xrightarrow{\mathrm{u.c.p.}}\int_0^tH_s\,dX_s$}
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>.
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