Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The interplay of classes of algorithmically random objects   We study algorithmically random closed subsets of $2^\omega$, algorithmically
random continuous functions from $2^\omega$ to $2^\omega$, and algorithmically
random Borel probability measures on $2^\omega$, especially the interplay
between these three classes of objects. Our main tools are preservation of
randomness and its converse, the no randomness ex nihilo principle, which say
together that given an almost-everywhere defined computable map between an
effectively compact probability space and an effective Polish space, a real is
Martin-L\"of random for the pushforward measure if and only if its preimage is
random with respect to the measure on the domain. These tools allow us to prove
new facts, some of which answer previously open questions, and reprove some
known results more simply.
  Our main results are the following. First we answer an open question of
Barmapalias, Brodhead, Cenzer, Remmel, and Weber by showing that
$\mathcal{X}\subseteq2^\omega$ is a random closed set if and only if it is the
set of zeros of a random continuous function on $2^\omega$. As a corollary we
obtain the result that the collection of random continuous functions on
$2^\omega$ is not closed under composition. Next, we construct a computable
measure $Q$ on the space of measures on $2^\omega$ such that
$\mathcal{X}\subseteq2^\omega$ is a random closed set if and only if
$\mathcal{X}$ is the support of a $Q$-random measure. We also establish a
correspondence between random closed sets and the random measures studied by
Culver in previous work. Lastly, we study the ranges of random continuous
functions, showing that the Lebesgue measure of the range of a random
continuous function is always contained in $(0,1)$.

20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
31
The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn