Lebesgue measure and integration, abstract measure spaces, product measures, Lp spaces, and probability built on top of them, through to the central limit theorem and martingales.
Measure-Theory
E. J. McShane, Integration, Princeton University Press, 1944 (Princeton Mathematical Series vol. 7). Lebesgue and Lebesgue-Stieltjes integration built from the Daniell point of view, written before measure theory settled into its modern textbook shape.
Worked solutions to all 203 exercises in Jeffrey S. Rosenthal's A First Look at Rigorous Probability Theory (2nd edition) — probability triples and extension theorems, random variables, expected values, inequalities and convergence, distributions, gambling games, discrete Markov chains, weak convergence, characteristic functions, decomposition of laws, conditional expectation, and martingales.
H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4th edition, Pearson,
- A MATH5825 companion text: Lebesgue measure and integration on the
line, differentiation, \(L^p\) spaces, metric and topological spaces, Banach and Hilbert space theory, and general measure and integration.
Solutions to every problem live at Solutions to Royden & Fitzpatrick’s Real Analysis.
Worked solutions to all 1062 problems in Royden & Fitzpatrick's Real Analysis (4th edition) — Lebesgue measure and integration for functions of a single real variable, differentiation and integration, the Lp spaces, metric and topological spaces, Banach and Hilbert spaces, and general measure and integration theory.
Terence Tao, An Introduction to Measure Theory, AMS GSM 126, 2011. The MATH5825 companion text. Tao develops the theory partly through the exercises themselves — Lebesgue measure from elementary measure, the integration convergence theorems, modes of convergence, differentiation theorems, product measures — with the exercises woven inline through the text rather than collected at section ends.
Solutions to every exercise live at Solutions to Tao’s An Introduction to Measure Theory.
Worked solutions to all 237 exercises in Terence Tao's An Introduction to Measure Theory (AMS GSM 126) — elementary and Lebesgue measure, the Lebesgue integral, abstract measure spaces, modes of convergence, differentiation theorems, outer measures and product measures, probability spaces, and the Kolmogorov extension theorem.
Worked solutions to all 587 exercises in Sheldon Axler's Measure, Integration & Real Analysis (Springer GTM 282) — Riemann integration, measures, integration, differentiation, product measures, Banach and Hilbert spaces, Lp spaces, real and complex measures, linear maps on Hilbert spaces, Fourier analysis, and probability measures.
Sheldon Axler, Measure, Integration & Real Analysis, Springer GTM 282, 2020. Open access — the free electronic edition lives at measure.axler.net.
Covers Riemann integration and its limitations, measures and their construction, integration, differentiation, product measures, Banach and Hilbert spaces, \(L^p\) spaces, real and complex measures, spectral theory for compact operators, and probability measures. Exercises sit at the end of each section (1A, 1B, …), with no printed answers.
Solutions to every exercise live at Solutions to Axler’s Measure, Integration & Real Analysis.
Worked solutions to all 523 exercises in Volume 1 of Bogachev's Measure Theory — constructions and extensions of measures, the Lebesgue integral, operations on measures and functions, the spaces Lp, and the integral and derivative.