  This paper presents a new observability estimate for parabolic equations in
$\Omega\times(0,T)$, where $\Omega$ is a convex domain. The observation region
is restricted over a product set of an open nonempty subset of $\Omega$ and a
subset of positive measure in $(0,T)$. This estimate is derived with the aid of
a quantitative unique continuation at one point in time. Applications to the
bang-bang property for norm and time optimal control problems are provided.
