With the usual path integral description, we have the formula $$\langle q''t''|q't'\rangle =\int\mathcal{D}q \exp{(iS)}$$ where $S=\int_{t'}^{t''}L(q,\dot{q})$ is the action evaluated for $t\in (t',t'').$ My question is about the insertion of a position operator $Q(t_1)$ for the propagator, i.e.
$$\langle q''t''|Q(t_1)|q't'\rangle=\int\mathcal{D}q \ q(t_1)\exp{(iS)}.$$
What does it meane to have a path integral with an integrand $q(t_1)$? After all, one way to interpret the path integral measure $\mathcal{D}q$ is to imagine an infinite number of slice of time intervals (and therefore slices of paths). In this case, should I understand $q(t_1)$ as the position operator at time $t_1$, or in the slice-view-point, position operator living in the time interval that contains $t_1$?
Or, is there a better way to understand this?