算法随机对象类的相互作用
The interplay of classes of algorithmically random objects
摘要：
我们研究算法随机闭子集的$2^ω$，算法随机连续函数从$2^ω$到$2^ω$，和算法随机Borel概率测度$2^ω$，特别是这三类对象之间的相互作用。我们的主要工具是保持随机性和它的逆，即无随机性无希洛原则，这两个原则一起说，给定有效紧概率空间和有效波兰空间之间的几乎处处定义的可计算映射，一个实数对于前推测度是马丁-洛夫随机的当且仅当它的原像对于定义域上的测度是随机的。这些工具使我们能够证明新的事实，其中一些回答了以前的开放问题，并更简单地重新证明了一些已知的结果。我们的主要结果如下。首先，我们回答一个开放的问题Barmapalias，布罗德黑德，岑策尔，Remmel，韦伯表明，$\mathcal{X}\subseteq2^ω$是一个随机闭集当且仅当它是一组零的随机连续函数的$2^ω$。作为推论，我们得到了$2^ω$上随机连续函数集在复合下不闭的结果.其次，我们在$2^ω$上的测度空间上构造了一个可计算测度$Q$，使得$\mathcal{X}\subseteq2^ω$是随机闭集当且仅当$\mathcal{X}$是$Q$ -随机测度的支集。我们还建立了随机闭集和卡尔弗在以前的工作中研究的随机措施之间的对应关系。最后，我们研究了随机连续函数的值域，证明了随机连续函数的值域的勒贝格测度总是包含在$(0,1)$中。
英文摘要：
We study algorithmically random closed subsets of $2^ω$, algorithmically random continuous functions from $2^ω$ to $2^ω$, and algorithmically random Borel probability measures on $2^ω$, 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-Löf 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 $\mathcal{X}\subseteq2^ω$ is a random closed set if and only if it is the set of zeros of a random continuous function on $2^ω$. As a corollary we obtain the result that the collection of random continuous functions on $2^ω$ is not closed under composition. Next, we construct a computable measure $Q$ on the space of measures on $2^ω$ such that $\mathcal{X}\subseteq2^ω$ 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 $(0,1)$.
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