From Jacod and Protter
9.5 Let be a probability space. Suppose that is a random variable with almost surely and . Define by . Show that defines a probability measure on .
9.7 Suppose that and let be defined as above. Let denote expectation with respect to . Show that .
Exercise 3. Suppose that and are random variables and that .
- Prove .
- If , the factor may be replaced with .
- If , the factor can be replaced with 1.
Exercise 4. Suppose that and . Prove for that
Hint: Think Cauchy-Schwarz.