  It was proved in 1998 by Ben-David and Litman that a concept space has a
sample compression scheme of size d if and only if every finite subspace has a
sample compression scheme of size d. In the compactness theorem, measurability
of the hypotheses of the created sample compression scheme is not guaranteed;
at the same time measurability of the hypotheses is a necessary condition for
learnability. In this thesis we discuss when a sample compression scheme,
created from com- pression schemes on finite subspaces via the compactness
theorem, have measurable hypotheses. We show that if X is a standard Borel
space with a d-maximum and universally separable concept class C, then (X,C)
has a sample compression scheme of size d with universally Borel measurable
hypotheses. Additionally we introduce a new variant of compression scheme
called a copy sample compression scheme.
