Let $X$ be a semimartingale und $H$ a predictable locally bounded process. For me $\int_0^t H_s dX_s$ was always just a notation for $(H \cdot X)_t$ and $H \cdot X$ is properly defined. On page 60 in the book Stochastic Integration and Differential Equations from Protter he writes
$$ (H \cdot X)_t=\int_0^t H_s dX_s=\int_{[0,t]} H_s dX_s$$
and defines
$$\int_{0+}^t H_s dX_s=\int_{(0,t]} H_s dX_s.$$
I don't understand the meaning of this. Can one define the stochastic integral on intervals?