25.1.1 Measures and distributionsE6CD9442A846

The representation of positive linear functionals by measures, theorem 2.3.3, is the Riesz representation theorem. For its formulation on locally compact, $\sigma $-compact spaces compare [Theorem 24, Tao09Radon]; on open subsets of Euclidean space our proof constructs the measure from a metric outer measure and identifies the functional by slicing functions into level sets. The regularity and density statements in proposition 2.2.2, proposition 2.3.5 correspond to [Theorem 21, Exercise 22 and Proposition 10, Tao09Radon]. For the measurability part of the outer-measure construction compare [Theorem 5, Tao10Products]. The assertions on product measures and the interchange of integrals in proposition 2.2.7 form the Tonelli–Fubini theorem for $\sigma $-finite measures; compare [Theorem 36 and its proof, Tao10Products], where the monotone class lemma (Lemma 35 there) is applied to the Boolean algebra of finite disjoint unions of rectangles. We instead apply Dynkin's $\pi $–$\lambda $ theorem, lemma 2.2.6, to the rectangles themselves.

For the local order bound, regularization by mollifiers and the fact that positive distributions are measures (definition 2.4.1, proposition 2.5.3, theorem 2.5.6) compare [Section 1, Exercises 4–5; Section 2, Definition 1, Exercise 15 and Lemma 2, Tao09Distributions]. That source writes the test function first and gives the space of distributions the conjugate complex structure. Our pairing $\langle u,\chi \rangle =\int u\chi \, \mathrm d x$ is complex linear in both variables, the convention used for currents later in the book; with this pairing the derivative of a distribution is defined with the same sign as in that source.

The selection theorem for sequences of measures, theorem 2.6.5, is proved by a diagonal argument over countably many test functions. The compactness theorem for nets, theorem 2.7.3, rests instead on Tychonoff's theorem for a product of intervals, which we prove with ultrafilters (lemma 2.7.2); this is where the axiom of choice enters. Subnets are Willard subnets throughout the book, and proposition 2.7.1 characterizes compactness in these terms. The Arzelà–Ascoli theorem, proposition 2.7.4, is proved in the usual way, by showing that an equicontinuous and pointwise bounded family is uniformly bounded and totally bounded for the uniform metric.

For the differentiation of integrals compare [Section 2, Theorems 32 and 36, Exercise 38 and Lemma 39, Tao10Differentiation]. In theorem 2.8.3 we use cubes instead of balls. The covering argument is the same: a finite disjoint subfamily of the cubes covers the original cubes after enlargement by the factor three, and the constant in the maximal inequality becomes $3^ d$. The Poincaré-type inequality lemma 2.8.4 follows by integrating the fundamental theorem of calculus along segments parallel to the coordinate axes.

For the Radon–Nikodym theorem compare [Corollary 1, following Theorem 2, Tao09RN]. The book uses only the case of a Borel measure that is dominated, at least locally, by a constant multiple of a Radon measure, lemma 2.8.1. We prove it by an $L^2$ argument on half-open dyadic cubes, so that no assumption on the measure of cube boundaries is needed. This lemma provides, for instance, the densities of the coefficient measures in lemma 7.14.3.