ALGORITHMICALLY RANDOM CLOSED SETS AND PROBABILITY A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Logan M. Axon Peter Cholak, Director Graduate Program in Mathematics Notre Dame, Indiana April 2010 ALGORITHMICALLY RANDOM CLOSED SETS AND PROBABILITY Abstract by Logan M. Axon Algorithmic randomness in the Cantor space, 2!, has recently become the subject of intense study. Originally defined in terms of the fair coin measure, algorithmic randomness has since been extended, for example in Reimann and Slaman [22, 23], to more general measures. Others have meanwhile developed definitions of algorithmic randomness for different spaces, for example the space of continuous functions on the unit interval (Fouch´e[8, 9]), more general topological spaces (Hertling and Weihrauch [12]), and the closed subsets of 2! (Barmpalias et al. [1], Kjos-Hanssen and Diamondstone [14]). Our work has also been to develop a definition of algorithmically random closed subsets. We take a very different approach, however, from that taken by Barmpalias et al. [1] and Kjos- Hanssen and Diamondstone [14]. One of the central definitions of algorithmic randomness in Cantor space is Martin-L¨ofrandomness. We use the probability theory of random closed sets (RACS) to prove that Martin-L¨ofrandomness can be defined in the space of closed subsets of any locally compact, Hausdorff, second countable space. We then explore the Martin-L¨ofrandom closed subsets of the spaces N, 2!, and R under different measures. In the case of 2! we prove that the definitions of Barm- palias et al. [1] and Kjos-Hanssen and Diamondstone [14] are compatible with Logan M. Axon our approach. In the case of N we prove that the Martin-L¨ofrandom subsets are exactly those with Martin-L¨ofrandom characteristic functions. In the case of R we investigate the Martin-L¨ofrandom closed sets under generalized Poisson processes. This leads to a characterization of the Martin-L¨ofrandom elements of R as exactly the reals contained in some Martin-L¨ofrandom closed subset of R. CONTENTS ACKNOWLEDGMENTS . iv CHAPTER 1: INTRODUCTION . .1 CHAPTER 2: BACKGROUND MATERIAL . .6 2.1 Notational conventions . .6 2.2 A (very) quick review of probability . .7 2.3 Algorithmic randomness . .9 2.4 Random closed sets . 14 2.4.1 Robbins' theorem . 20 2.4.2 The Choquet capacity theorem . 24 2.4.3 Examples of Choquet capacities . 31 2.5 Fractal geometry . 37 CHAPTER 3: MARTIN-LOF¨ RANDOM CLOSED SETS . 41 3.1 Random subsets of the natural numbers . 54 CHAPTER 4: MARTIN-LOF¨ RANDOM CLOSED SUBSETS OF CAN- TOR SPACE . 57 4.1 BBCDW-random closed sets . 57 4.1.1 Fractal properties . 66 4.2 Galton-Watson random closed sets . 70 4.3 Generalized random fractal constructions . 83 4.4 Random intervals . 90 4.5 Maxitive capacities . 94 4.6 Random singletons . 96 CHAPTER 5: MARTIN-LOF¨ RANDOM CLOSED SUBSETS OF THE REAL NUMBERS UNDER GENERALIZED POISSON PROCESSES 98 5.1 A generalized Poisson process . 98 5.2 More on generalized Poisson processes . 115 ii REFERENCES . 127 iii ACKNOWLEDGMENTS This dissertation would not have existed without the help and support of many people. My adviser, Peter Cholak, got me going first place and then kept me going with advice, encouragement, and when necessary, deadlines. Bjørn Kjos-Hanssen also played a big role by asking the right questions and pointing me in the right direction on many occasions. Julia Knight, Dan Mauldin, Andrew Sommese, and Sergei Starchenko also helped by listening and reading with critical ears and eyes. My office-mates, Chris, Sean, Josh, and Steve, saw everything in its earliest form and had many helpful ideas and suggestions. Thanks to all of you. And thanks to Bonnie. iv CHAPTER 1 INTRODUCTION It is easy to generate a random binary sequence: simply flip a fair coin and record the outcome of each flip as 0 for heads and 1 for tails. The laws of prob- ability tell us that in the long run we should expect to get the same number of 0s and 1s, that we can expect to see any finite sequence of 0s and 1s, et cetera. What we certainly do not expect to get is a sequence like 10110111011110 ::: , or any other sequence with a pattern. However, this sequence is as likely to be the outcome of our coin flips as any other sequence (each infinite sequence occurs with probability zero). Of course most sequences look more random than the sequence 10110111011110 ::: , even if each one is just as likely to occur. Algorith- mic randomness (also called effective randomness) formalizes the idea that some sequences are more random than others by combining computability theory with probability. One of the most important definitions in algorithmic randomness is Martin-L¨of randomness which we introduce in section 2.3. Algorithmically random sequences have received a lot of attention recently. Many of the results in the area of algorith- mically random sequences have been collected in Downey et al. [5] and Nies [20]. The explosion of interest in algorithmically random sequences has led researchers to ask whether similarly interesting behavior is possible in other settings. One alternative setting is C[0; 1], the set of continuous functions from the unit interval 1 to R. This was first studied by Fouch´e,[9] and [8], and later by Kjos-Hanssen and Nerode [15] who showed that results about algorithmically random functions could be obtained by producing a measure algebra homomorphism between 2! and C[0; 1]. As we will see this basic idea can be used in other settings, in particular in the space of closed subsets of certain topological spaces. Algorithmically random closed sets were first studied by Barmpalias et al. [1] who defined a notion of randomness for closed subsets of Cantor space by coding each infinite binary tree without dead ends as a ternary real. Every closed subset of Cantor space can be uniquely represented as the set of paths through such a tree. Barmpalias et al. [1] defined a closed set to be random if the code for the corresponding tree was Martin-L¨ofrandom. This definition is explored here in section 4.1. Probability theorists and statisticians, on the other hand, have defined a ran- dom closed set to be something quite different. A random closed set as defined in the literature (in particular the books of Matheron [16], Molchanov [18], and Nguyen [19]) is simply a closed set-valued random variable. That is, a random closed set is a measurable map from a probability space to F(E), the space of closed sets the topological space E. This is formalized using the hit-or-miss topol- ogy (also known as the Fell topology) on the space F(E). We introduce this probabilistic theory of random closed sets in section 2.4. This is obviously a very different idea than that developed by Barmpalias et al. [1] but there are connec- tions. We prove that the coding of closed sets of Cantor space used by Barmpalias et al. [1] is actually an example of a particularly nice random closed set (in the probability theory sense). This is lemma 4.1.4 in section 4.1. As in the case of real-valued random variables, random closed sets (in the 2 probability sense) induce an probability measure on the target space. The connec- tion with algorithmic randomness comes when we use such a measure to produce Martin-L¨oftests in the space of closed sets. This allows for the study Martin-L¨of random closed sets purely from the perspective of the space of closed sets and the hit-or-miss topology. In section 3 we develop a general theory of Martin-L¨of random closed sets. One key result here is proposition 3.0.9, which establishes that Martin-L¨ofrandomness can be defined in the space of closed subsets of a lo- cally compact, Hausdorff, second countable space using the hit-or-miss topology. This allows us to talk about Martin-L¨ofrandom closed subsets of such spaces. We note that different random closed sets give rise to different measures and hence different classes of Martin-L¨ofrandom closed sets. Other important results in this section are the technical lemmas 3.0.12 and 3.0.13, which are used extensively in later examples. The bulk of this paper is an exploration of examples of specific random closed sets (in the probability sense) and the Martin-L¨ofrandom closed sets they give rise to. We begin by looking at the example of Martin-L¨ofrandom closed subsets of N. In section 3.1 we prove that the Martin-L¨ofrandom closed subsets of N are exactly those subsets with a Martin-L¨ofrandom characteristic function (in 2!). Our first major example of a random closed set is the coding defined by Barm- palias et al. [1]. Having established in lemma 4.1.4 that this coding is a measurable map from 3! !F(2!) we are then able to prove that a closed set is Martin-L¨of random in F(2!) if and only if it is the image of a Martin-L¨ofrandom element of 3! (corollary 4.1.5). This means that our definition of Martin-L¨ofrandom closed sets agrees exactly with the definition of algorithmically random closed sets given by Barmpalias et al. [1]. Our approach, however, allows for the use of theorems from 3 probability theory of random closed sets. In proposition 4.1.9, for instance, we use one of these tools (Robbins' theorem) to prove that for this example Martin-L¨of random closed sets have measure 0.