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Register a new userEffective notions of weak convergence of measures on the real line
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We establish a framework for the study of the effective theory of weak convergence of measures. We define two effective notions of weak convergence of measures on $mathbb{R}$: one uniform and one non-uniform. We show that these notions are equivalent. By means of this equivalence, we prove an effective version of the Portmanteau Theorem, which consists of multiple equivalent definitions of weak convergence of measures.
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In this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence (as shown by Kautz) and every 2-random sequence forms a minimal pair in the Turing degrees with every 2-generic sequence (as shown by Nies, Stephan, and Terwijn). We strengthen these results by showing that every Demuth random sequence computes a 1-generic sequence (which answers an open question posed by Barmpalias, Day, and Lewis) and that every Demuth random sequence forms a minimal pair with every pb-generic sequence (where pb-genericity is an effective notion of genericity that is strictly between 1-genericity and 2-genericity). Moreover, we prove that for every comeager $mathcal{G}subseteq 2^omega$, there is some weakly 2-random sequence $X$ that computes some $Yinmathcal{G}$, a result that allows us to provide a fairly complete classification as to how various notions of effective randomness interact in the Turing degrees with various notions of effective genericity.
We study the question, ``For which reals $x$ does there exist a measure $mu$ such that $x$ is random relative to $mu$? We show that for every nonrecursive $x$, there is a measure which makes $x$ random without concentrating on $x$. We give several conditions on $x$ equivalent to there being continuous measure which makes $x$ random. We show that for all but countably many reals $x$ these conditions apply, so there is a continuous measure which makes $x$ random. There is a meta-mathematical aspect of this investigation. As one requires higher arithmetic levels in the degree of randomness, one must make use of more iterates of the power set of the continuum to show that for all but countably many $x$s there is a continuous $mu$ which makes $x$ random to that degree.
We investigate which infinite binary sequences (reals) are effectively random with respect to some continuous (i.e., non-atomic) probability measure. We prove that for every n, all but countably many reals are n-random for such a measure, where n indicates the arithmetical complexity of the Martin-Lof tests allowed. The proof is based on a Borel determinacy argument and presupposes the existence of infinitely many iterates of the power set of the natural numbers. In the second part of the paper we present a metamathematical analysis showing that this assumption is indeed necessary. More precisely, there exists a computable function G such that, for any n, the statement `All but countably many reals are G(n)-random with respect to a continuous probability measure cannot be proved in $ZFC^-_n$. Here $ZFC^-_n$ stands for Zermelo-Fraenkel set theory with the Axiom of Choice, where the Power Set Axiom is replaced by the existence of n-many iterates of the power set of the natural numbers. The proof of the latter fact rests on a very general obstruction to randomness, namely the presence of an internal definability structure.
We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and several other problems related to computing isomorphisms. These other problems include Scott analysis (in the form of back-and-forth relations), jump hierarchies, and computing descending sequences in linear orders.
A {it weak selection} on $mathbb{R}$ is a function $f: [mathbb{R}]^2 to mathbb{R}$ such that $f({x,y}) in {x,y}$ for each ${x,y} in [mathbb{R}]^2$. In this article, we continue with the study (which was initiated in cite{ag}) of the outer measures $lambda_f$ on the real line $mathbb{R}$ defined by weak selections $f$. One of the main results is to show that $CH$ is equivalent to the existence of a weak selection $f$ for which: [ mathcal lambda_f(A)= begin{cases} 0 & text{if $|A| leq omega$,} infty & text{otherwise.} end{cases} ] Some conditions are given for a $sigma$-ideal of $mathbb{R}$ in order to be exactly the family $mathcal{N}_f$ of $lambda_f$-null subsets for some weak selection $f$. It is shown that there are $2^mathfrak{c}$ pairwise distinct ideals on $mathbb{R}$ of the form $mathcal{N}_f$, where $f$ is a weak selection. Also we prove that Martin Axiom implies the existence of a weak selection $f$ such that $mathcal{N}_f$ is exactly the $sigma$-ideal of meager subsets of $mathbb{R}$. Finally, we shall study pairs of weak selections which are almost equal but they have different families of $lambda_f$-measurable sets.
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