  This paper proposes a new notion of typical sequences on a wide class of
abstract alphabets (so-called standard Borel spaces), which is based on
approximations of memoryless sources by empirical distributions uniformly over
a class of measurable "test functions." In the finite-alphabet case, we can
take all uniformly bounded functions and recover the usual notion of strong
typicality (or typicality under the total variation distance). For a general
alphabet, however, this function class turns out to be too large, and must be
restricted. With this in mind, we define typicality with respect to any
Glivenko-Cantelli function class (i.e., a function class that admits a Uniform
Law of Large Numbers) and demonstrate its power by giving simple derivations of
the fundamental limits on the achievable rates in several source coding
scenarios, in which the relevant operational criteria pertain to reproducing
empirical averages of a general-alphabet stationary memoryless source with
respect to a suitable function class.
