This uncommonly fine textbook of the modern theory of functions of a real variable is particularly well-suited for mathematically mature students in the fields of philosophy and foundations of mathematics, philosophy of physics, probability theory, and statistics. Those who wish to achieve first-hand acquaintance with the quantum theory will also need to have a grasp of the material presented in this book. The first chapter presents a capsule survey of topics in abstract set theory and algebra, including a discussion of (...) relations, functions, the axiom of choice, and the well-ordering theorem, and leading up to the construction of the fields of real and complex numbers. The second chapter is an introduction to topology and spaces of topological functions. The approach is set-theoretical. Chapter three deals with the theory of the Lebesgue integral and measure theory. This chapter is quite comprehensive, rigorous and embellished with many exercises for the student. Subsequent sections include the theories of function spaces, Banach spaces, abstract Hilbert spaces, and differentiation. This is clearly one of the more worthwhile texts in its field.—H. P. K. (shrink)

The Jordan-Hahn decomposition and the Lebesgue decomposition, two basic notions of classical measure theory, are generalized for measures on orthomodular posets. The Jordan-Hahn decomposition property (JHDP) and the Lebesgue decomposition property (LDP) are defined for sections Δ of probability measures on an orthomodular poset L. If L is finite, then these properties can be characterized geometrically in terms of two parallelity relations defined on the set of faces of Δ. A section Δ is shown to have the JHDP (...) if and only if every pair of f-parallel faces is p-parallel; it is shown to have the LDP if and only if every pair of disjoint faces is p-parallel. It follows from these results that the LDP is stronger than the JHDP in the setting of finite orthomodular posets. Mielnik's convex scheme of quantum theory provides the frame for a physical interpretation of these results. (shrink)

We use the compactness theorem of continuous logic to give a new proof that $$L^r([0,1]; {\mathbb {R}})$$ isometrically embeds into $$L^p([0,1]; {\mathbb {R}})$$ whenever $$1 \le p \le r \le 2$$. We will also give a proof for the complex case. This will involve a new characterization of complex $$L^p$$ spaces based on Banach lattices.

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of Lp-computable functions with a size notion, by introducing Lp-computable Baire categories. We show that Lp-computable Baire categories satisfy the following three basic properties. Singleton sets {f } are meager, suitable infinite unions of meager sets are meager, and the whole space of Lp-computable functions is not meager. We give an alternative characterization of meager sets via Banach-Mazur games. We (...) study the convergence of Fourier series for Lp-computable functions and show that whereas for every p > 1, the Fourier series of every Lp-computable function f converges to f in the Lp norm, the set of L1-computable functions whose Fourier series does not diverge almost everywhere is meager. (shrink)

Universality of generalized Alexandroff's cube plays essential role in theory of absolute retracts for the category of, -closure spaces. Alexandroff's cube. is an, -closure space generated by the family of all complete filters. in a lattice of all subsets of a set of power.Condition P(,, ) says that is a closure space of all, -filters in the lattice ( ).

In standard probability theory, events of probability zero may still occur. That is, a real number $x$ can be sampled uniformly at random from $[0,1]$ despite having $\mathbb{P}(X=x)=0$. This goes against the intuitive principle that probability $0$ means impossible. In this paper, we make this tension explicit and argue that the usual resolution is backwards. We begin by analyzing two common claims: 1. (Zero) Probability $0$ means impossible for any possible outcome. 2. (Uniform) It is possible to sample a real (...) number: there exists an experiment whose possible outcomes are all $x\in[0,1]$, distributed according to the uniform (Lebesgue) measure, and which returns a completed real on each run. These two claims are incompatible. Standard probability theory keeps (Uniform) and abandons (Zero), allowing events of probability zero to occur. We argue for the opposite choice. We retain (Zero) and deny that there is any literal experiment that samples a completed real from $[0,1]$ with the uniform law. The main tool is definability. We propose that to count as a number at all, an object must be definable by a finite description in a fixed language. Undefinable ``real numbers'' then do not exist as numbers; they are set theoretic artifacts rather than genuine numerical quantities. Under any continuous distribution on $[0,1]$, the set of definable reals has probability zero, so an idealized ``random real'' is almost surely an undefinable pseudo object. On our view this is a sign that the usual continuum picture is being applied outside its proper domain. We develop this finite information perspective across several domains: definable reals and complex numbers, definable infinitesimals and infinite quantities in hyperreal fields, definable distributions such as the Dirac delta, and definable infinities in extended number systems. We stress the abundance of definable numerical objects: giving up undefinable reals does not impoverish the numerical universe but focuses attention on a rich and extensible world of definable numbers. We also revisit Cantor's diagonal argument. We accept its conclusion as a theorem of set theory about the uncountability of the set theoretic reals, but argue that its anti diagonal construction presupposes the existence of undefinable real codes. Once we restrict attention to definable numbers, Cantor's theorem is about a larger background universe than the domain of genuine numerical quantities. Finally, we articulate a process based view of sampling on the continuum. No physical or computational procedure ever produces an infinite real; what exists in practice is a sampling process that yields arbitrarily fine but always finite information. In this operational setting, events with probability zero that refer to completed infinite outcomes never occur, and the principle that probability $0$ means impossible holds for all physically meaningful events. (shrink)

Probabilism holds that rational credence functions are probability functions defined over some probability space $(\Omega, \F, P)$. According to some recent philosophical arguments, in some situations, rational credence function must be \textit{total}, i.e. $\F=2^\Omega$, a view which I call \textit{credence totalism}. Arguments for credence totalism are based on the premise that non-Lebesgue measurable subsets of $\mathbb{R}$ are epistemically significant, in the sense that an agent has reasons to assign probability to these sets. This paper argues that nonmeasurable sets are (...) not epistemically significant in this sense. Consequently, the arguments for credence totalism are not successful. My argument is based on a careful consideration of the role of the Axiom of Choice in probabilistic practice. I also discuss some topics considered closely related, viz. the existence of total chance functions and the truth value of the Continuum Hypothesis. I argue that the role of nonmeasurability in epistemology does not shed light on these issues. (shrink)

Gyenis and Rédei (G&R) have shown that any prior p on a finite algebra A, however chosen, significantly restricts the set of posteriors derivable from p by Jeffrey conditioning (JC) on a nontrivial measurable partition (i.e., a partition consisting of members of A, at least one of which is not an atom of A). They support this claim by proving that the set of potential posteriors not derivable from p in this way, which they call the Bayes blind spot of (...) p, is large, having cardinality c and normalized Lebesgue measure 1, as well as being of second Baire category for a natural metrizable topology. In the present paper, we establish results analogous to those of G&R for probability measures on any infinite sigma algebra of subsets of a countably infinite set (which requires distinctly different treatments of the topological and measure-theoretic cases). We also show, in both the finite and infinite cases, that all of the limitative results for a single prior p continue to hold for the intersection of the Bayes blind spots of countably many priors. This leads us to reject the claim of G&R that the large size of blind spots in the single prior case is attributable to the limitations imposed by priors. We argue instead that it is the so-called rigidity property of JC that accounts for the large size of Bayes blind spots. G&R also prove that any potential posterior q can be derived from a prior p by at most two applications of JC on nontrivial partitions. But they remark that their particular two-stage derivation of an $$r \in BS(p)$$ from p is barely distinguishable from a single, complete reassessment on a trivial partition. We show, however, that there are two-stage derivations of r from p that require no direct assessment of the probability of any atom at either stage. Finally, in order to situate the aforementioned limitative results in the proper context, we demonstrate that a probability revision that would amount to complete reassessment within a Jeffrey evidentiary framework may arise, in a different evidentiary framework, in a way that involves no direct assessment of the probability of any atom, as, for example, in a generalization of Jeffrey’s solution to the problem of old evidence and new explanation. (shrink)

We prove that the property "P doesn't make the old reals Lebesgue null" is preserved under countable support iterations of proper forcings, under the additional assumption that the forcings are nep (a generalization of Suslin proper) in an absolute way. We also give some results for general Suslin ccc ideals.

We introduce the notion of effective Axiom A and use it to show that some popular tree forcings are Suslin+. We introduce transitive nep and present a simplified version of Shelah’s “preserving a little implies preserving much”: If I is a Suslin ccc ideal (e.g. Lebesgue-null or meager) and P is a transitive nep forcing (e.g. P is Suslin+) and P does not make any I-positive Borel set small, then P does not make any I-positive set small.

Let ω be a Kolmogorov–Chaitin random sequence with ω1: n denoting the first n digits of ω. Let P be a recursive predicate defined on all finite binary strings such that the Lebesgue measure of the set {ω|∃nP(ω1: n )} is a computable real α. Roughly, P holds with computable probability for a random infinite sequence. Then there is an algorithm which on input indices for any such P and α finds an n such that P holds within the (...) first n digits of ω or not in ω at all. We apply the result to the halting probability Ω and show that various generalizations of the result fail. (shrink)

A theory of approximation to measurable sets and measurable functions based on the concepts of recursion theory and discrete complexity theory is developed. The approximation method uses a model of oracle Turing machines, and so the computational complexity may be defined in a natural way. This complexity measure may be viewed as a formulation of the average-case complexity of real functions—in contrast to the more restrictive worst-case complexity. The relationship between these two complexity measures is further studied and compared with (...) the notion of the distribution-free probabilistic computation. The computational complexity of the Lebesgue integral of polynomial-time approximable functions is studied and related to the question “FP = ♯P?”. (shrink)

This paper investigates aspects of measure and randomness in the context of locale theory . We prove that every measure μ, on the σ-frame of opens of a fitted σ-locale X, extends to a measure on the lattice of all σ-sublocales of X . Furthermore, when μ is a finite measure with μ=M, the σ-locale X has a smallest σ-sublocale of measure M . In particular, when μ is a probability measure, X has a smallest σ-sublocale of measure 1. All (...) σ prefixes can be dropped from these statements whenever X is a strongly Lindelöf locale, as is the case in the following applications. When μ is the Lebesgue measure on the Euclidean space Rn, Theorem 1 produces an isometry-invariant measure that, via the inclusion of the powerset P in the lattice of sublocales, assigns a weight to every subset of Rn. When μ is the uniform probability measure on Cantor space {0,1}ω, the smallest measure-1 sublocale, given by Theorem 2, provides a canonical locale of random sequences, where randomness means that all probabilistic laws are satisfied. (shrink)

Concepts of L1 space, integrable functions and integrals are formalized in weak subsystems of second order arithmetic. They are discussed especially in relation with the combinatorial principle WWKL (weak-weak König's lemma and arithmetical comprehension. Lebesgue dominated convergence theorem is proved to be equivalent to arithmetical comprehension. A weak version of Lebesgue monotone convergence theorem is proved to be equivalent to weak-weak König's lemma.

We propose a reformulation of the ideal $\mathcal {N}$ of Lebesgue measure zero sets of reals modulo an ideal J on $\omega $, which we denote by $\mathcal {N}_J$. In the same way, we reformulate the ideal $\mathcal {E}$ generated by $F_\sigma $ measure zero sets of reals modulo J, which we denote by $\mathcal {N}^*_J$. We show that these are $\sigma $ -ideals and that $\mathcal {N}_J=\mathcal {N}$ iff J has the Baire property, which in turn is equivalent (...) to $\mathcal {N}^*_J=\mathcal {E}$. Moreover, we prove that $\mathcal {N}_J$ does not contain co-meager sets and $\mathcal {N}^*_J$ contains non-meager sets when J does not have the Baire property. We also prove a deep connection between these ideals modulo J and the notion of nearly coherence of filters (or ideals).We also study the cardinal characteristics associated with $\mathcal {N}_J$ and $\mathcal {N}^*_J$. We show their position with respect to Cichoń’s diagram and prove consistency results in connection with other very classical cardinal characteristics of the continuum, leaving just very few open questions. To achieve this, we discovered a new characterization of $\mathrm {add}(\mathcal {N})$ and $\mathrm {cof}(\mathcal {N})$. We also show that, in Cohen model, we can obtain many different values to the cardinal characteristics associated with our new ideals. (shrink)

We study Lebesgue and Atsuji spaces within subsystems of second order arithmetic. The former spaces are those such that every open covering has a Lebesgue number, while the latter are those such that every continuous function defined on them is uniformly continuous. The main results we obtain are the following: the statement “every compact space is Lebesgue” is equivalent to $\hbox{\sf WKL}_0$ ; the statements “every perfect Lebesgue space is compact” and “every perfect Atsuji space is (...) compact” are equivalent to $\hbox{\sf ACA}_0$ ; the statement “every Lebesgue space is Atsuji” is provable in $\hbox{\sf RCA}_0$ ; the statement “every Atsuji space is Lebesgue” is provable in $\hbox{\sf ACA}_0$ . We also prove that the statement “the distance from a closed set is a continuous function” is equivalent to $\Pi^1_1\hbox{-\sf CA}_0$. (shrink)

Analyzing the effective content of the Lebesgue density theorem played a crucial role in some recent developments in algorithmic randomness, namely, the solutions of the ML-covering and ML-cupping problems. Two new classes of reals emerged from this inquiry: thepositive density pointswith respect toeffectively closed sets of reals, and a proper subclass, thedensity-one points. Bienvenu, Hölzl, Miller, and Nies have shown that the Martin-Löf random positive density points are exactly the ones that do not compute the halting problem. Treating this (...) theorem as our starting point, we present several new results that shed light on how density, randomness, and computational strength interact. (shrink)

A textbook for the undergraduate who is meeting the Lebesgue integral for the first time, relating it to the calculus and exploring its properties before deducing the consequent notions of measurable functions and measure.

In this work we give a complete answer as to the possible implications between some natural properties of Lebesgue measure and the Baire property. For this we prove general preservation theorems for forcing notions. Thus we answer a decade-old problem of J. Baumgartner and answer the last three open questions of the Kunen-Miller chart about measure and category. Explicitly, in \S1: (i) We prove that if we add a Laver real, then the old reals have outer measure one. (ii) (...) We prove a preservation theorem for countable-support forcing notions, and using this theorem we prove (iii) If we add ω 2 Laver reals, then the old reals have outer measure one. From this we obtain (iv) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg B(m) + \neg U(m) + U(c))$ . In \S2: (i) We prove a preservation theorem, for the finite support forcing notion, of the property " $F \subseteq ^\omega\omega$ is an unbounded family." (ii) We introduce a new forcing notion making the old reals a meager set but the old members of ω ω remain an unbounded family. Using this we prove (iii) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + U(m) + \neg B(c) + \neg U(c) + C(c))$ . In \S3: (i) We prove a preservation theorem, for the finite support forcing notion, of a property which implies "the union of the old measure zero sets is not a measure zero set," and using this theorem we prove (ii) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg U(m) + C(m) + \neg C(c))$. (shrink)

Festschrift honoring P.F. Strawson; includes contributed articles on his contributions in logic and on logic.

We prove that consistently the Lebesgue null ideal is not Tukey reducible to the Silver null ideal. This contrasts with the situation for the meager ideal which, by a recent result of the author, Spinas [Silver trees and Cohen reals, Israel J. Math. 211 473–480] is Tukey reducible to the Silver ideal.

Pv~P: Cambridge Journal of Undergraduate Philosophy, Issue 1, 1982.

This is a well-behaved concept in Newtonian physics. But a center of gravity is not an atom or a subatomic particle or any other physical item in the world. It has no mass; it has no color; it has no physical properties at all, except for spatio-temporal location. It is a fine example of what Hans Reichenbach would call an abstractum. It is a purely abstract object. It is, if you like, a theorist's fiction. It is not one of the (...) real things in the universe in addition to the atoms. But it is a fiction that has nicely defined, well delineated and well behaved role within physics. (shrink)