  The John-Nirenberg spaces $JN_p$ are generalizations of the space of bounded
mean oscillation $BMO$ with $JN_{\infty}=BMO$. Their vanishing subspaces
$VJN_p$ and $CJN_p$ are defined in similar ways as $VMO$ and $CMO$, which are
subspaces of $BMO$. As our main result, we prove that $VJN_p$ and $CJN_p$
coincide by showing that certain Morrey type integrals of $JN_p$ functions tend
to zero for small and large cubes. We also show that $JN_{p,q}(\mathbb{R}^n) =
L^p(\mathbb{R}^n) / \mathbb{R}$, if $p = q$.
