These days, studying probability without measure is like studying physics without calculus. If you have done some baby measure theory in stage 3, you are probably ready for the followings.
Check out Probability Theory As Extended Logic for collected probability papers.
Probaility built upon Measure Theory:
- Ash R.B. and Doleans-Dade C.A. Probability & Measure Theory
- Athreya K.B. and Lahiri S.N. Measure Theory and Probability Theory
- Bass R.F. Probability Theory [.pdf] (FREE!)
- Billingsley P. Convergence of Probability Measures
- Billingsley P. Probability and Measure
- Bobrowski A. Functional Analysis for Probability and Stochastic Processes: An Introduction
- Breiman L. Probability
- Capinski M. and Kopp P.E. Measure, Integral and Probability
- Chung K.L. A Course in Probability Theory
- Dembo A. Probability Theory [.pdf] (FREE!)
- Dudley R.M. Real Analysis and Probability
- Feller W. An Introduction to Probability Theory and Its Applications Vol. 2
- Geiss C. and Geiss S. An Introduction to Probability Theory [.pdf] (FREE!)
- Gnedenko B.V. and Kolmogorov A.N. Limit Distributions for Sums of Independent Random Variables
- Gray R.M. Probability, Random Processes, and Ergodic Properties (FREE!)
- Gut A. Probability: A Graduate Course
- Jaynes E.T. Probability Theory: The Logic of Science (here for FREE!)
- Jorgensen P.E.T. Analysis and Probability: Wavelets, Signals, Fractals - Meant for wide range of audiences, nice historical background, remarks and references.
- Kallenberg O. Foundations of Modern Probability - Nice historical and bibliographical notes are included at the back.
- Knill O. Probability and Stochastic Processes with Applications [.pdf] (FREE!)
- Loève M. Probability Theory I, II - Fairly comprehensive.
- Kolmogorov A.N. Foundations of the Theory of Probability (FREE!) - For historical interest.
- Shiryaev A.N. Probability - The material is based on a three-semester course of lectures, under the title Probability, Statistics, Stochastic Processes, I, II, given by the author in the Moscow State University. He draws nice comparison between set-theoretic interpretation and interpretation in probability theory, see p.136-137.
- Strasser H. Introduction to Probability Theory and Stochastic Processes [.pdf] (FREE!)
- Tsirelson B. Advanced Probability Theory (FREE!)
- Wilde I.F. Measures, Integration and Probability (FREE!)
- Williams D. Probability with Martingales
- Brzezniak Z. and Zastawniak T. Basic Stochastic Processes
- Doob J.L. Stochastic Processes - Considered to be classics in the field.
- Dembo A. Stochastic Processes [.pdf] (FREE!)
- Jacod J. and Shiryaev A.N. Limit Theorems for Stochastic Processes
- Karlin S. and Taylor H.M. A Second Course in Stochastic Processes - Amazon user Yan Zhang comments that this is a must-have if you wanna go to Wall Street.
- Mörters P. and Peres Y. Brownian Motion [.pdf] (FREE!)
- Parzen E. Stochastic Processes
- Pollard D. Convergence of Stochastic Processes (here for FREE!)
- Ross S.M. Stochastic Processes - See also Russell Lyons' course notes [.pdf].
- Shalizi S. Stochastic Processes (FREE!)
- Todorovic P. An Introduction to Stochastic Processes and Their Application
- van Zanten H. An Introduction to Stochastic Processes in Continuous Time [.pdf] (FREE!)
- Varadhan S.R.S. Stochastic Processes (FREE!)
- Bain A. Stochastic Calculus (FREE!)
- Bichteler, K. Stochastic Integration and Stochastic Differential Equations (FREE!)
- Chung K.L. and Williams R.J. An Introduction to Stochastic Integration
- Evans L.C. An Introduction to Stochastic Differential Equations [.pdf] (FREE!) - Evans wrote a popular book on PDE.
- Geiss S. Scripts (FREE!) - Stochastic processes, DE, and models.
- Karatzas I. and Shreve S.E. Brownian Motion and Stochastic Calculus
- Klebaner F.C. Introduction to Stochastic Calculus with Applications
- Kurtz T.G. Lectures on Stochastic Analysis [.pdf] (FREE!)
- Malliavin P. Stochastic Analysis
- Mörters P. Lecture Notes on Stochastic Analysis [.ps] (FREE!)
- Seppalainen T. Basics of Stochastic Analysis (FREE!)
- Varadhan S.R.S. Stochastic Analysis (FREE!)
- Wilde I.F. Stochastic Analysis (FREE!)