  Suppose a $k$-uniform hypergraph $H$ that satisfies a certain regularity
instance (that is, there is a partition of $H$ given by the hypergraph
regularity lemma into a bounded number of quasirandom subhypergraphs of
prescribed densities). We prove that with high probability a large enough
uniform random sample of the vertex set of $H$ also admits the same regularity
instance. Here the crucial feature is that the error term measuring the
quasirandomness of the subhypergraphs requires only an arbitrarily small
additive correction. This has applications to combinatorial property testing.
The graph case of the sampling result was proved by Alon, Fischer, Newman and
Shapira.
