  This paper presents two observability inequalities for the heat equation over
$\Omega\times(0,T)$. In the first one, the observation is from a subset of
positive measure in $\Omega\times(0,T)$, while in the second, the observation
is from a subset of positive surface measure in $\partial\Omega \times(0,T)$.
It also proves the Lebeau-Robbiano spectral inequality when $\Omega$ is a
bounded Lipschitz and locally star-shaped domain. Some applications for the
above-mentioned observability inequalities are provided.
