  In this paper we develop a measure-theoretic method to treat problems in
hypergraph theory. Our central theorem is a correspondence principle between
three objects: An increasing hypergraph sequence, a measurable set in an
ultraproduct space and a measurable set in a finite dimensional Lebesgue space.
Using this correspondence principle we build up the theory of dense hypergraphs
from scratch. Along these lines we give new proofs for the Hypergraph Removal
Lemma, the Hypergraph Regularity Lemma, the Counting Lemma and the Testability
of Hereditary Hypergraph Properties. We prove various new results including a
strengthening of the Regularity Lemma and an Inverse Counting Lemma. We also
prove the equivalence of various notions for convergence of hypergraphs and we
construct limit objects for such sequences. We prove that the limit objects are
unique up to a certain family of measure preserving transformations. As our
main tool we study the integral and measure theory on the ultraproduct of
finite measure spaces which is interesting on its own right.
