  We prove a removal lemma for induced ordered hypergraphs, simultaneously
generalizing Alon--Ben-Eliezer--Fischer's removal lemma for ordered graphs and
the induced hypergraph removal lemma. That is, we show that if an ordered
hypergraph $(V,G,<)$ has few induced copies of a small ordered hypergraph
$(W,H,\prec)$ then there is a small modification $G'$ so that $(V,G',<)$ has no
induced copies of $(W,H,\prec)$. (Note that we do \emph{not} need to modify the
ordering $<$.)
  We give our proof in the setting of an ultraproduct (that is, a Keisler
graded probability space), where we can give an abstract formulation of
hypergraph removal in terms of sequences of $\sigma$-algebras. We then show
that ordered hypergraphs can be viewed as hypergraphs where we view the
intervals as an additional notion of a ``very structured'' set. Along the way
we give an explicit construction of the bijection between the ultraproduct
limit object and the corresponding hyerpgraphon.
