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.
Analysis
N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland, 1958 (Bibliotheca Mathematica vol. IV). The classic short course on asymptotic expansions: the O and o calculus, implicit functions, summation by Euler’s formula, Laplace’s method, saddle points, the method of stationary phase, iterated functions and differential equations. A 1958 hardcover copy.
Worked solutions to all 467 exercises in Stephen Abbott's Understanding Analysis (2nd edition) — the real numbers and completeness, sequences and series, the topology of R, functional limits and continuity, the derivative, sequences and series of functions, the Riemann integral, and the additional topics chapter from metric spaces to Fourier series.
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.
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.
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.