Self-study: what fractions of problems to solve? I am self-studying measure-theoretic probability out of Billingsley's Probability and Measure. So far I have been trying to solve all the exercises. While the exercises are wonderful and I can ultimately solve most they do take up a lot of time. 
I would like to know what fraction of problems from this or similar books are students expected to solve in graduate courses on this topic?
 A: Billingsley's Probability and Measure is a very comprehensive book. Usually the subjects of the book are covered in two separate graduate courses, the one being "Theoretical Probability" and the other being "Stochastic Processes" (or a similar course title). Graduate students are usually expected to solve 5 to 10 exercises on each chapter but it is also expected - in addition to that - that they are able to understand and reproduce the basic proofs (understanding of main ideas and and ability to identify and use the mathematical "tools" that are used for the proof). So, for a self-learner I would suggest the following


*

*Separate the book in two subjects, one about theoretical probability (Chapters 1, 2, 3, partly 4, 5 and 6) and one about markov chains, stochastic processes and martingales (chapters 7 and partly 4, 5 and 6).

*Try to reproduce the main proofs (or most of the proofs) since in the proofs are applied (or demonstrated) many interesting and useful techniques. Usually it is difficult for a person to think of the techniques by himself, but once he has grasped them by learning the proofs, they can help him substantially to reach an advanced level (as is probably your goal, if I understand correctly).

*Start solving the exercises, that have already hints or solutions at the back of the book. (usually these are the odd numbered exercises, but in this book that is not the case).


Coping firstly with the above tasks has the advantage that you have the solutions and you can advance faster (by gaining on the same time many useful knowledge) without wasting time on problems that you do not know whether you are on the correct path or not. If you are so apt to tackle the above, then proceed also to the unsolved exercises. In that case your abilities and comprehension of the subject will be at a higher level than that of the average graduate students. 
A: Solving problems does take time, but it is usually time well-spent. You can learn a fair amount by watching the author develop the material, and by filling in details in the text. But, you learn far more by doing as many exercises as you can. Exercises develop technique, knowledge of and facility with the theorems and definitions, and give you the opportunity to solidify the material in your mind. You really don't get these benefits any other way. As former American Secretary of State Colin Powell said: "There are no secrets to success. It is the result of preparation, hard work, and learning from failure." So, get working on those exercises. Do as many as you can, and ask for help on the rest.
P.S. My remarks are predicated on the assumption you really want to learn this material. If you want only to be acquainted with it, you can get by with a lesser investment of time, but you still will need to do a fair number of problems (perhaps half of them, or, maybe fewer).
A: No royal road to geometry, as the ancient greeks used to say.
Unfortunately, the same holds even today for measure theory and measure theoretic probability.
If this stuff is new to you, the more exercises you solve the better and deeper you understand the subject. (In particular, if this is the first Measure Theory class you take.) Billingsley's Probability and Measure is a great textbook, and the more time you spend on it the better your investment.
When I was your age I always had as a target to solve every single exercise. However, one thing your do not specify is how much of the book is covered in your class. Apparently, it is impossible to cover this book in a semester. Are you doiing only the probability chapters/sections or the whole thing including stochastic processes?
