  This paper proposes a systematic mathematical analysis of both the direct and
inverse acoustic scattering problem given the source in Radon measure space.
For the direct problem, we investigate the well-posedness including the
existence, the uniqueness, and the stability by introducing a special
definition of the weak solution, i.e. \emph{very} weak solution. For the
inverse problem, we choose the Radon measure space instead of the popular $L^1$
space to build the sparse reconstruction, which can guarantee the existence of
the reconstructed solution. The sparse reconstruction problem can be solved by
the semismooth Newton method in the dual space. Numerical examples are
included.
