Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Best approximation properties in spaces of measurable functions   We research proximinality of $\mu$-sequentially compact sets and
$\mu$-compact sets in measurable function spaces. Next we show a correspondence
between the Kadec-Klee property for convergence in measure and
$\mu$-compactness of the sets in Banach function spaces. Also the property $S$
is investigated in Fr\'echet spaces and employed to provide the Kadec-Klee
property for local convergence in measure. We discuss complete criteria for
continuity of metric projection in Fr\'echet spaces with respect to the
Hausdorff distance. Finally, we present the necessary and sufficient condition
for continuous metric selection onto a one-dimensional subspace in sequence
Lorentz spaces $d(w,1)$.

29
The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph
21
The categories of this paper are: adap-org
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
20
The categories of this paper are: acc-phys
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs