The rank of the sparse signals brought by multiple measurement vectors (MMV) augments the performance of joint sparse recovery. In general, suppose the sparsity level <inline-formula><tex-math notation="LaTeX">$ k$</tex-math></inline-formula> is less than or equal to <inline-formula><tex-math notation="LaTeX">$ [rank(\boldsymbol{X})+spark(\boldsymbol{A})-1]/2$</tex-math></inline-formula>, the sparsest solution of the MMV problem is unique and recoverable via various methods. It is shown in this letter that the unique solution of the sparsity level <inline-formula><tex-math notation="LaTeX">$ k$</tex-math></inline-formula> up to <inline-formula><tex-math notation="LaTeX">$ spark(\boldsymbol{A})-1$</tex-math></inline-formula> actually exists in a measure theoretical point of view. More specifically, even when <inline-formula><tex-math notation="LaTeX">$ [rank(\boldsymbol{X})+spark(\boldsymbol{A})-1]/2\leq k< spark(\boldsymbol{A})$</tex-math></inline-formula>, the sparsest solution to <inline-formula><tex-math notation="LaTeX">$ \boldsymbol{A}\boldsymbol{X}=\boldsymbol{Y}$</tex-math></inline-formula> is still unique with full Lebesgue measure in every <inline-formula><tex-math notation="LaTeX">$ k$</tex-math></inline-formula>-sparse coordinate space. This phenomenon is fully confirmed by the MMV tail-<inline-formula><tex-math notation="LaTeX">$ \ell _{2,1}$</tex-math></inline-formula> minimization technique. Furthermore, the phenomenon that the traditional <inline-formula><tex-math notation="LaTeX">$ \ell _{2,1}$</tex-math></inline-formula> minimization actually fails to recover <inline-formula><tex-math notation="LaTeX">$ \boldsymbol{X}$</tex-math></inline-formula> with <inline-formula><tex-math notation="LaTeX">$ k \geq [spark(\boldsymbol{A})-1]/2$</tex-math></inline-formula> is investigated from the same perspective of measure theory. Extensive numerical tests conducted by the MMV tail-<inline-formula><tex-math notation="LaTeX">$ \ell _{2,1}$</tex-math></inline-formula> minimization and <inline-formula><tex-math notation="LaTeX">$ \ell _{2,1}$</tex-math></inline-formula> minimization are demonstrated to confirm the findings. The tail minimization procedure exhibits the most prominent effectiveness for the larger sparsity levels among all known techniques.