  Let $X,Y$ be Banach spaces, $(S_t)_{t \geq 0}$ a $C_0$-semigroup on $X$, $-A$
the corresponding infinitesimal generator on $X$, $C$ a bounded linear operator
from $X$ to $Y$, and $T > 0$. We consider the system \[ \dot{x}(t) = -Ax(t),
\quad y(t) = Cx(t) \quad t\in (0,T], \quad x(0) = x_0 \in X. \] We provide
sufficient conditions such that this system satisfies a final state
observability estimate in $L_r ((0,T) ; Y)$, $r \in [1,\infty]$. These
sufficient conditions are given by an uncertainty relation and a dissipation
estimate. Our approach unifies and generalizes the respective advantages from
earlier results obtained in the context of Hilbert spaces. As an application we
consider the example where $A$ is an elliptic operator in $L_p(\mathbb{R}^d)$
for $1<p<\infty$, and where $C = \mathbf{1}_\omega$ is the restriction onto a
thick set $\omega \subset \mathbb{R}^d$. In this case, we show that the above
system satisfies a final state observability estimate if and only if $\omega
\subset \mathbb{R}^d$ is a thick set. Finally, we make use of the well-known
relation between observability and null-controllability of the predual system,
and investigate bounds on the corresponding control costs.
