Measure

Many of the inequalities that we shall establish originally concern finite sequences and finite sums. We then extend them to infinite sequences and infinite sums, and to functions and integrals, and it is these more general results that are useful in applications.

Although the applications can be useful in simple settings – concerning the Riemann integral of a continuous function, for example – the extensions are usually made by a limiting process. For this reason we need to work in the more general setting of measure theory, where appropriate limit theorems hold. We give a brief account of what we need to know; the details of the theory will not be needed, although it is hoped that the results that we eventually establish will encourage the reader to master them. If you are not familiar with measure theory, read through this chapter quickly, and then come back to it when you find that the need arises.

Suppose that Ω is a set. A measure ascribes a size to some of the subsets of Ω. It turns out that we usually cannot do this in a sensible way for all the subsets of Ω, and have to restrict attention to the measurable subsets of Ω. These are the ‘good’ subsets of Ω, and include all the sets that we meet in practice. The collection of measurable sets has a rich enough structure that we can carry out countable limiting operations.

Decreasing rearrangements

Suppose that (E, ∥.∥E ) is a Banach function space and that f ∈ E. Then ∥f∥E = ∥|f|∥E , so that the norm of f depends only on the absolute values of f. For many important function spaces we can say more. Suppose for example that f ∈ Lp , where 1 < p < ∞. By Proposition 1.3.4, ∥f∥p = (p ∫ t p−1µ(|f| > t)dt)1/p , and so ∥f∥p depends only on the distribution of |f|. The same is true for functions in Orlicz spaces. In this chapter, we shall consider properties of functions and spaces of functions with this property.

In order to avoid some technical difficulties which have little real interest, we shall restrict our attention to two cases:

Convex sets and convex functions

Many important inequalities depend upon convexity. In this chapter, we shall establish Jensen's inequality, the most fundamental of these inequalities, in various forms.

A subset C of a real or complex vector space E is convex if whenever x and y are in C and 0 ≤ θ ≤ 1 then (1 − θ)x + θy ∈ C. This says that the real line segment [x, y] is contained in C. Convexity is a real property: in the complex case, we are restricting attention to the underlying real space. Convexity is an affine property, but we shall restrict our attention to vector spaces rather than to affine spaces.

Proposition 4.1.1A subset C of a vector space E is convex if and only if whenever x 1, …, xn ∈ C and p 1, …, pn are positive numbers with p 1 + … + pn = 1 then p 1 x 1 + … + pnxn ∈ C.

Proof The condition is certainly sufficient. We prove necessity by induction on n. The result is trivially true when n = 1, and is true for n = 2, as this reduces to the definition of convexity. Suppose that the result is true for n − 1, and that x 1, …, xn and p 1, …, pn are as above.

Inequalities lie at the heart of a great deal of mathematics. G.H. Hardy reported Harald Bohr as saying ‘all analysts spend half their time hunting through the literature for inequalities which they want to use but cannot prove’. Inequalities provide control, to enable results to be proved. They also impose constraints; for example, Gromov's theorem on the symplectic embedding of a sphere in a cylinder establishes an inequality that says that the radius of the cylinder cannot be too small. Similar inequalities occur elsewhere, for example in theoretical physics, where the uncertainty principle (which is an inequality) and Bell's inequality impose constraints, and, more classically, in thermodynamics, where the second law provides a fundamental inequality concerning entropy.

Thus there are very many important inequalities. This book is not intended to be a compendium of these; instead, it provides an introduction to a selection of inequalities, not including any of those mentioned above. The inequalities that we consider have a common theme; they relate to problems in real analysis, and more particularly to problems in linear analysis. Incidentally, they include many of the inequalities considered in the fascinating and ground-breaking book Inequalities, by Hardy, Littlewood and Pólya, originally published in 1934.

The first intention of this book, then, is to establish fundamental inequalities in this area. But more importantly, its purpose is to put them in context, and to show how useful they are.