Group-norms are vector-space norms but with the scalars restricted to units (invertibles), ±1. The Birkhoff–Kakutani theorem (a first-countable Hausdorff topological group has a right-invariant metric) we view as a normability theorem rather than a metrization theorem, a relative of Kolmogorov’s normability theorem for topological vector spaces (the condition for whose normability is that the origin have a convex bounded neighbourhood). The groups here need not be abelian, so one has left-sided and right-sided versions. Proved here is the Analytic Baire Theorem: if a normed group contains an (either-sided) non-meagre analytic set, it is Baire, separable and (modulo a meagre set) itself analytic. Other results here include the ‘Analytic Shift Theorem’ and the ‘Squared Pettis Theorem’, category relatives of the Steinhaus Difference Theorem.

The infinite combinatorics developed in the previous chapters may be harnessed to give a treatment of regular variation in quite general contexts. Particularly useful tools here are the Category Embedding Theorem and the Effros Theorem. The main theorems of regular variation (see, e.g., BGT) include the Uniform Convergence Theorem (UCT) and the Characterization Theorem. The UCT is extended to the $L_1$-algebra of a locally compact metric group, using Reiter-like conditions from amenability. The Characterization Theorem can be formulated for normed groups X and H, with T a connected non-meagre Baire subgroup of the group of homeomorphisms from X to H. If for h : X → H is Baire and $h(tx)h(x)^{-1} \rightarrow k(t)$ for x → ∞ in X, then k is a continuous homomorphism from T to H. A calculus of regular variation is developed, involving the ‘differential modulus’. The theory is extended to the case of non-commutative H.

The Category Embedding Theorem (CET) is a result in infinite combinatorics related to the Kestelman–Borwein–Ditor Theorem KBD, and also to the concept of shift-compactness. The relationships between KBD, CET and various forms of No Trumps NT are given.

The text proper of the book begins with Littlewood’s three principles. The first – ‘any measurable set is nearly a finite union of intervals’ – is essentially regularity of Lebesgue measure. The second – ‘any measurable function is nearly continuous’ – is Lusin’s Theorem. The third – ‘any convergent sequence of measurable functions is nearly uniformly continuous’ – is Egorov’s Theorem. Then what will be needed from general topology is summarised, with references, going as far as para-compactness. Modes of convergence – in measure (in probability), almost everywhere (almost sure), etc. – are discussed. The Borel hierarchy – the result of applying, to (say) the open sets, the sigma and delta operations (union and intersection) alternately – is developed, as far as the Souslin operation. Analytic sets – much used in the book – are briefly treated here.

As a counterpart to Chapter 9 on category–measure duality, we focus here on a variety of situations in which duality fails. We cover a range of topics: Liouville numbers, Banach–Tarski (‘paradoxical’) decompositions, restriction and continuity, random series, normal numbers, topological and Hausdorff dimension, random Dirichlet series, filters, genericity, the Fubini and Kuratowski–Ulam theorems. We give an account of modern results on forcing, deferring technicalities to Chapter 16.

The other prime example of a fine topology is the fine topology of potential theory (in the usual sense of electromagnetism, gravitation, etc.) This is finer than the Euclidean topology but coarser than the density topology. Each of these three topologies has its σ-ideal of small sets: the meagre sets for the Euclidean case, the polar sets for the fine topology of potential theory, and the (Lebesgue-)null sets for the density topology. The polar sets have been extensively studied, not only in potential theory as above but in probabilistic potential theory; pioneers here include P.-A. Meyer and J. L. Doob. Relevant here are the links between martingales and harmonic functions (likewise their sub- and super-versions), Green functions, Green domains, Markov processes, Brownian motion, Dirichlet forms, energy and capacity. The general theory of such fine topologies involves such things as analytically heavy topologies, base operators, density operators and lifting.

Steinhaus’ Theorem of Chapter 9, an interior-point result, was extended from the line under Lebesgue measure to topological groups under Haar measure by Weil. The resulting Steinhaus–Weil theory, which is extensive, is presented in Chapter 15. The Simmons–Mospan converse gives the condition for the extension to hold: in a locally compact Polish group, a Borel measure has the ‘Steinhaus–Weil property’ if and only if it is absolutely continuous with respect to Haar measure. We define measure subcontinuity (adapted from Fuller’s subcontinuity for functions), and amenability at the identity. We prove Solecki’s interior-point theorem: in a Polish group, if a set E is not left Haar null, then the identity is an interior point of $E^{-1}E$. Related results on sets such as ${AB}^{-1}$ rather than ${AA}^{-1}$ are given.

A point is a density point of a set if the ratio of the length of its intersection with an interval containing it to that of the interval tends to 1 as the interval shrinks to the point. The classical Lebesgue Density Theorem states that almost all points of a measurable set are density points. Declaring a set open when all its points are density points leads to a topology, the density topology. This is a fine topology – it refines the ordinary (Euclidean) topology, in having more open sets. The density-meagre sets are the Lebesgue-null sets. This result shows how working bitopologically – switching between the Euclidean and density topologies – enables us to switch between the category and measure cases. A list of properties of the line under the density topology is given. Caution is needed: for instance, the line is a topological group under the Euclidean topology, but not (only a paratopological group) under the density topology (as now multiplication is only separately but not jointly continuous).

We begin with the canonical status of the reals: this extends up to uniqueness to within isomorphism as a complete Archimedean ordered field, but not up to cardinality aspects. We discuss four ‘elephants in the room’ here (an elephant in the room is something obviously there but which no one wants to mention). The first elephant (from Gödel’s incompleteness theorem and the Continuum Hypothesis, CH): one cannot properly speak of the real line, but rather which real line one chooses to work with. The second is ‘which sets of reals can one use?’ (it depends on what axioms of set theory one assumes – in particular, the role of the Axiom of Choice, AC). The third is that there are sentences that are neither provable nor disprovable, and that no non-trivial axiom system is capable of proving its own consistency. Thus, we do not – cannot – know that mathematics itself is consistent. The fourth elephant is that even to define cardinals, the concept of cardinality needs AC.

The close parallels – duality – between category and measure go back to Sierpiński in the 1920s. The Sierpiński–Erdős duality principle (see Oxtoby, Ch. 19) gives full duality under the Continuum Hypothesis, CH (see Chapter 16). An unconditional example of duality is the Poincaré recurrence theorem from statistical mechanics. So is the zero–one law of probability theory; duality does not extend as far as its quantitative versions, the strong law of large numbers and the ergodic theorem. Also discussed are: uniqueness theorems for trigonometric series; the Cauchy functional equation; the Steinhaus dichotomy – for S Baire and non-meagre (or measurable and non-null), the difference set S - S contains an interval around the origin (the dichotomy being that this may fail for meagre/null sets). Combinatorial principles are relevant here – Jensen’s Diamond and Ostaszewski’s Club are here augmented by a third, No Trumps, NT, a common generalization of the Baire and measurable cases.

Part of the motivation for this book was its role in solving open problems in regular variation – in brief, the study of limiting relations of the form f(λx)/f(x) → g(x) as x → ∞ for all λ > 0 and its relatives. This was the subject of the earlier book Regular Variation by N. H. Bingham, C. M. Goldie and J. L. Teugels (BGT). So to serve as prologue to the present book, a brief summary of the many uses of regular variation is included, to remind readers of BGT and spare others needing to consult it. Topics covered include: probability (weak law of large numbers, central limit theorem, stability, domains of attraction, etc.), complex analysis (Abelian, Tauberian and Mercerian theorems, Levin–Pfluger theory), analytic number theory (prime divisor functions; results of Hardy and Ramanujan, Erdős and Kac, Rényi and Turán); the Cauchy functional equation g(λμ) = g(λ)g(μ) for all λ; μ > 0; dichotomy – the solutions are either very nice (powers) or very nasty (pathological – the ‘Hamel pathology’).

The Open Mapping Theorem – that a surjective continuous linear map between Fréchet spaces is open – is one of the cornerstones of linear functional analysis. Effros’ Theorem is a group-action counterpart: a continuous transitive action by a Polish group G on a non-meagre space X is open (i.e. point-evaluation maps $g \mapsto gx$ are open). We discuss action, micro-transitivity and shift-compactness. We investigate what we call the crimping property (arising in Banach’s classic book of 1932). It turns out that this is equivalent to the Effros property. Various related results are proved.