First off I must say we haven't had a publication in measure theory or abstract probability for decades which integrates as much specialty knowledge and wide range of application as Pollard's 2002 "A User's Guide to Measure Theoretic Probability" that is able to prove it! Previous to this work, all these unneccesary distinctions and misunderstandings have been made (and are still being made) between the discrete and the continuous in mathematics, and physics as well. Im not going to spoil the suprises on how it's done but will simply point out that this work should soon be prerequisite reading for all graduates moving on towards pure mathematics and general-unified field theoretic applications. Once we can get a concrete understanding of this work we may soon no longer teach nor practice probability theory and mathematics as separated theories nor as separated fields!
A User's Guide to Measure Theoretic Probability is a quality book, as are all the books in the Cambridge Series in Statistical and Probabilistic Mathematics (see Wavelet Methods for Time Series Analysis, the Determination and Tracking of Frequency, Bayesian Methods). Illustrations are included in the book as well. You can have a look at the book in PDF format on Pollard's website
Topical contents of interest for this book include:
Reveals that independence of random variables by means of distribution functions can be done metricaly using product measures instead of factorizing joint densities and assuming independence as transformational smoothness. In other words you can actually do the math smoothly instead of generalizing it as such.
The discrete and the continuous no longer have to be taught at the graduate level as though they were differential. This fact is proven in Chapter III by means of derivation theory, decomposition (not loss but preservation/re-composition) of Lebesgue measure.
Univariate and multivariate no longer distinct? Once again the proofs are in the book. Distributions and joint distributions using grainy calculations, Jacobians, many integrals, and matrices, are easily achieved independantly (mind you with a bit of extra intuitive rigour) using measure theory. Distributions can be determined as image measures and joint distributions as image measures for mappings into product spaces discussed in Chapters II and IV.
The first two chapters introduce this beneficial application in probability with:
Measures and sigma-fields, Measureable functions, Integrals, Construction of integrals from measures, Limit theorems, Neglible sets, Lp-spaces, Uniform integrability, Image measures and distributions, Generating classes of sets and functions,
Chapter III is about Desnsities and Derivatives:
Densities and absolute continuity, The Lebesgue decomposition, Distances and affinities between measures, Classical concepts of absolute continuity, Vitali covering lemma, etc.
Chapter IV Product Spaces and Independence: Independence of sigma-fields, Construction of measures on product spaces, Product measures, Infinite product spaces etc.
there are 351 pages and chapters 1-4 only go to page 108. The rest of the book includes massive ammounts of data on: Conditional distributions, Integration and disintregration, Conditional densities, Invariance, Kolmogorov's abstact conditional expectation, Sufficiency. Chapter VI is all about Martingales; Stopping times, Convergence, Krickeberg decomposition, Uniform integrability, Reversed Martingales, Symmetry, Exchangeability, Lindeberg's method, Multivariate limit theorems, Fourier transforms, Martingale central limit theorem, the Levy and Cramer-Wold theorems, Brownian motion etc.
Also by Chapter 10 you get into: Representations and Couplings;
Strassen's theorem, the Yurinskii coupling, Quantile coupling of Binomials with normals, Haar coupling etc. Chapters 11 and 12: Exponential Tails and and the Law of the Iterated Logarithm; Identically distributed summands, Multivariate Normal Distributions; Ferniques inequality, and it's proof, Gaussian isoperimetric inequality, and it's prrof. The extensive Appendices include topics: Measures and inner measures, Tightness, Countable additivity, Extension to the ^c-closure, Integral representations, Hilbert Spaces; Orthogonal projections, Orthonormal bases, Series expansions, Convexity; Convex sets and functions, their Integral representations, their relative interiors, Quantile coupling of the Binomial with the normal, Martingales in Ccontinuous Time; using filtrations, Brownian filtrations, supermartingales etc. Disintegration of Measures; Representations of measures on product spaces, disintegrations with respect to a measurable map etc.
Most of the material covered has previously only been available in French contexts and we are lucky to have now in english, available to Phd's in the United States and written at the Graduate level for students worldwide.