Math 714 Midterm Review
The following should help prepare for the midterm
that will given on April 6.
The midterm covers chapters 4, 5 and sections 9.1 though 9.4.
The midterm does not cover Section 4.7 or
Theorem 9.12 from Section 9.4.
However, it may include some simple surprise problems
in addition to what is on this list.
Definitions
- Definitions 4.1, 4.2, 4.3.
- The completion of a sigma algebra given in Theorem 4.2.
- Definitions 4.10, 4.11, 4.15, 4.16, 4.17, 4.18, 4.22, 4.23, 4.24.
- Definitions 5.1, 5.3, 5.4, 5.5, 5.6, 5.8, 5.10, 5.11, 5.12.
- Definitions 9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9.
Statements of Theorems
- Monotone Convergence Theorem (Theorem 4.?).
- Fatou's Lemma (Theorem 4.7).
- Dominated Convergence (Theorem 4.9).
- Borel-Cantelli Lemma (Theorem 5.?).
- Strong Law of Large Numbers (Theorem 5.8, Colorally 5.4).
- Holder's Inequality (Theorem 9.9).
- Minkowski's Inequality (Theorem 9.10).
Homework Problems
- All problems from assignments 1, 2 and 4.
Proofs of Theorems
- Part (a) of Borel-Cantelli Lemma.
- Theorem 5.6, Proposition 5.8.
- Proof that exp(-x) >= 1-x for x >= 0.
- Proof that |ab| < (1/p)|a|^p +
(1/q)|b|^1 when 1/p + 1/q = 1.
- Proposition 9.3.
- The proofs of Proposition 9.4 and Theorem 9.6 presented in class.
- Theorem 9.9 and Theorem 9.10.
- Proof that L^1 is complete as in Theorem 9.11 with p = 1.