About This Course

This book builds the theory of measure and the Lebesgue integral from the ground up and carries it as far as the theorems that the rest of analysis leans on every day: monotone and dominated convergence, Tonelli and Fubini, completeness of the Lebesgue spaces, the Radon–Nikodym theorem, and the Lebesgue differentiation theorem. It is written for a reader working alone at the level of a first graduate course. Every main theorem is proved in full, and every hypothesis is tested against an example in which the conclusion fails without it.

You need a first proof-based course in analysis. From Real Analysis: suprema and infima, convergence of sequences and series, compactness of closed bounded intervals, uniform convergence, and the Riemann integral. From set theory: unions, intersections, complements, images and preimages, countable and uncountable sets. A little linear algebra, the idea of a vector space and of a norm, is used from Chapter 7 on. When a fact from another course is needed it is stated with the hypotheses the argument uses and marked once with its library identifier; it is not proved again here.

Six abilities mark the reader who has finished the book, and each should be available with the book closed. Decide whether a family of sets is a \(\sigma\) -algebra and whether a set function on it is a measure, and extend a length-like function from intervals to a complete measure. Prove that a function is measurable and approximate it by simple functions. Compute a Lebesgue integral and move a limit, a sum, or a derivative through it, naming the theorem that permits the move and the function that verifies its hypothesis. Change the order of an iterated integral, or show by example why the change is illegitimate. Estimate products and sums in the \(L^p\) norms and prove that a Cauchy sequence there has a limit. Decompose a signed measure, find a density of one measure with respect to another, and recover a function from its local averages.

Some conventions are fixed throughout. The natural numbers are \(\mathbb N=\{1,2,3,\dots\}\) , and countable means finite or countably infinite. For subsets of a fixed set \(X\) we write \(A^c=X\setminus A\) for the complement, \(A\setminus B=A\cap B^c\) , and \(A\triangle B=(A\setminus B)\cup(B\setminus A)\) . The indicator function \(\mathbf 1_A\) equals \(1\) on \(A\) and \(0\) off it. A sequence of sets increases to \(A\) , written \(A_n\uparrow A\) , when \(A_1\subseteq A_2\subseteq\cdots\) and \(\bigcup_n A_n=A\) ; it decreases to \(A\) , written \(A_n\downarrow A\) , when \(A_1\supseteq A_2\supseteq\cdots\) and \(\bigcap_n A_n=A\) . Measures take values in \([0,\infty]\) , where \(a+\infty=\infty\) for every \(a\) , where \(a\cdot\infty=\infty\) for \(a>0\) , and where \(0\cdot\infty=0\) . The last convention is the one that makes the integral of the zero function over a set of infinite measure equal to zero, and it is used without comment. From Chapter 2 on, \(\lambda\) always denotes Lebesgue measure.

The exercises that close each chapter begin with recall and end with proofs. Work them with pencil before opening the companion solutions, which give complete arguments for some, graduated hints for others, and bare answers for a few. In a proof course a full solution is a written proof, and you should hold your own work to that standard: every limit interchange justified by a named theorem, every exceptional set identified. Several exercises present a plausible argument and ask for the flaw. Take those seriously, because nearly every error in applied analysis is an unjustified interchange of two limiting operations, and the only protection is the habit of asking which hypothesis licenses each step. The cumulative review at the end mixes the chapters without saying which theorem each problem needs.

Two neighbouring subjects are deliberately left alone. The probabilistic reading of this material, in which a measure of total mass one is a probability and a measurable function is a random variable, together with independence, conditional expectation, and the limit laws, belongs to Measure-Theoretic Probability. The abstract theory of Banach and Hilbert spaces and of the operators between them belongs to Functional Analysis. This book supplies the spaces and the convergence theorems that both of them start from.