Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Families of well approximable measures   We provide an algorithm to approximate a finitely supported discrete measure
$\mu$ by a measure $\nu_{N}$ corresponding to a set of $N$ points so that the
total variation between $\mu$ and $\nu_N$ has an upper bound. As a consequence
if $\mu$ is a (finite or infinitely supported) discrete probability measure on
$[0,1]^{d}$ with a sufficient decay rate on the weights of each point, then
$\mu$ can be approximated by $\nu_N$ with total variation, and hence
star-discrepancy, bounded above by $(\log N) N^{-1}$. Our result improves, in
the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that
for any normalized Borel measure $\mu$, there exist finite sets whose
star-discrepancy with respect to $\mu$ is at most $(\log N)^{d-\frac{1}{2}}
N^{-1}$. Moreover we close a gap in the literature for discrepancy in the case
$d=1$ showing both that Lebesgue is indeed the hardest measure to approximate
by finite sets and also that all measures without discrete components have the
same order of discrepancy as the Lebesgue measure.

29
The categories of this paper are: funct-an
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
26
The categories of this paper are: bayes-an
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom