  Solving compressed sensing problems relies on the properties of sparse
signals. It is commonly assumed that the sparsity s needs to be less than one
half of the spark of the sensing matrix A, and then the unique sparsest
solution exists, and recoverable by $\ell_1$-minimization or related
procedures. We discover, however, a measure theoretical uniqueness exists for
nearly spark-level sparsity from compressed measurements Ax = b. Specifically,
suppose A is of full spark with m rows, and suppose $\frac{m}{2}$ < s < m. Then
the solution to Ax = b is unique for x with $\|x\|_0 \leq s$ up to a set of
measure 0 in every s-sparse plane. This phenomenon is observed and confirmed by
an $\ell_1$-tail minimization procedure, which recovers sparse signals uniquely
with s > $\frac{m}{2}$ in thousands and thousands of random tests. We further
show instead that the mere $\ell_1$-minimization would actually fail if s >
$\frac{m}{2}$ even from the same measure theoretical point of view.
