"Show that if (X \in L^1) and (\mathcalG) is a sub-(\sigma)-algebra, then (|E[X|\mathcalG]| \le E[|X| | \mathcalG])."
If you can produce a self-contained solution to Exercise 14.5 without peeking, you have mastered the book. You no longer need a solution manual—you are the solution manual. David Williams Probability With Martingales Solutions
In the book, Williams provides a detailed introduction to martingales, including their definition, properties, and applications. He also discusses various types of martingales, such as discrete-time martingales, continuous-time martingales, and local martingales. "Show that if (X \in L^1) and (\mathcalG)