Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniform Approximation of Vapnik-Chervonenkis Classes   For any family of measurable sets in a probability space, we show that either
(i) the family has infinite Vapnik-Chervonenkis (VC) dimension or (ii) for
every epsilon > 0 there is a finite partition pi such the pi-boundary of each
set has measure at most epsilon. Immediate corollaries include the fact that a
family with finite VC dimension has finite bracketing numbers, and satisfies
uniform laws of large numbers for every ergodic process. From these
corollaries, we derive analogous results for VC major and VC graph families of
functions.

22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
33
The categories of this paper are: plasm-ph