The tightness hypothesis of the Fr'echet--Kolmogorov criterion cannot be dropped

By [F1] and [F2], $\mathcal{F}$ is bounded, ${\sup }_k\mathrm{\parallel }{u}_k{\mathrm{\parallel }}_{L^p}=\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^p}<\mathrm{\infty }$, and ${\sup }_k\mathrm{\parallel }{\tau }_h{u}_k-u_k{\mathrm{\parallel }}_{L^p}=\mathrm{\parallel }{\tau }_h\phi -\phi {\mathrm{\parallel }}_{L^p}\to 0$ as $\mathrm{\mid }h\mathrm{\mid }\to 0$: the family is uniformly translation continuous.

Fix $R>0$ and choose a radius $A>0$ with $\mathrm{supp}\phi \subseteq B(0,A)$ and then an integer $k>R+A$; then up to a null set the support of $u_k$ lies outside $B(0,R)$, so ${\int }_{\mathrm{\mid }x\mathrm{\mid }>R}\mathrm{\mid }{u}_k{\mathrm{\mid }}^{p}dx=\mathrm{\parallel }{u}_k{\mathrm{\parallel }}_{L^p}^p=\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^p}^p>0$, and this value is independent of $R$; hence no $R$ makes the tails uniformly small, and the tightness condition of [F4] fails.

By [F3] the family is not relatively compact, so — although it is bounded and uniformly translation continuous — it violates the conclusion of the criterion [F4]; the two remaining hypotheses do not force compactness, and the refuted claim is false. Countable and Dependent Choice are used only through the criterion [F4] and its negated instance; the Sobolev and translation suppliers also use Countable Choice.