Back to MAT324 Homepage
### MAT 324: Final Exam 8:00am-10:45am in Physics P-124.

### Topics Covered

These will be the chapters covered from the textbook:

Marek Capinski & Ekkehard Kopp, Measure, Integral and Probability, 2nd ed., Springer-Verlag, Springer Undergraduate Mathematics Series, ISBN 1-85233-781-8.

**Chapter 1:** Cantor's diagonal Argument, Riemann Integration.

**Chapter 2:** Null Sets, Cantor Set, Outer Measure (+ properties), Lebesgue Measure (+ properties), Sigma Fields, Borel Measure, Completions of Measure spaces, Probability Spaces.

**Chapter 3:** Measurable Functions, Properties of Measurable Functions (including examples, limits etc), Random Variables, Probability Distributions, Indpendence of Random Variables. We will not cover chapter 3.5.5 on mathematical finance).

**Chapter 4:** Lebesgue Integral of simple functions and of non-negative measurable functions, some basic properties of these integrals, Fatou's lemma (with examples), monotone convergence theorem (MCT) with examples and the proof of additivity of integral using MCT. Integrable functions along with some basic properties, dominated convergence theorem (DCT), examples where DCT holds and does not hold, Beppo-Levi theorem, use of DCT and Beppo-Levi to compute integrals. Comparison of Riemann and Lebesgue integral, approximation of measurable functions by continuous and step functions in the L^{1} sense and the Riemann-Lebesgue theorem. Probability distributions+ proof of uniqueness, cumulative distribution functions, expectation of a random variable, characteristic function. Again we will not cover chapter 4.7.5 on mathematical finance.

**Chapter 5:** Normed vector spaces, completeness, L^{1} (+ proof that it is a complete normed vector space), L^{2}, Schwarz Inequality, Inner prodct spaces, Hilbert Spaces and proof that L^{2} is a Hilbert space. Parallelogram Law and orthogonal projections. L^{p} spaces, HÃ¶lder's inequality, Minkowski's inequality, proof that L^{p} is a normed vector space. Relationships between L^{p} and L^{q} for p not equal to q and examples. We will not cover Section 5.4 on probability.

**Chapter 6:** Instead of Sections 6.1-6.4, we covered material from
here. Here we covered the definition of product measures, monotone classes, construction of the product measure, Fubini's theorem, examples where Fubini's theorem holds and also does not hold. We also explained how Fubini's theorem holds for completions of product measures. We also covered Section 6.5.1 on joint probability distributions and joint density functions. Section 6.5.2 on indpendence was covered and conditional probability in Section 6.5.3 was also covered. We did not cover conditional expectation from Section 6.5.3. We also did not cover Sections 6.5.4 and 6.5.5 and so these Sections will not be examined.

**Chapter 7:** Section 7.1 on densities and conditioning and Section 7.2 the Radon-Nikodym theorem and the Lebesgue decomposition theorem.
You will not need to know the contents of chapters 7.3, 7.4, 7.5 and chapter 8.

**Student Accessibility Support Center (SASC) Statement:**

If you have a physical, psychological, medical or learning disability that may impact your course work, please contact the Student Accessibility Support Center (SASC), ECC (Educational Communications Center) Building, room 128, (631) 632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential.

Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and the staff at the Student Accessibility Support Center (SASC). For procedures and information go to the following website:
http://www.stonybrook.edu/ehs/fire/disabilities.
**Academic Integrity Statement:**

Each student must pursue his or her academic goals
honestly and be personally accountable for all submitted work. Representing another
person's work as your own is always wrong. Faculty are required to report any suspected
instances of academic dishonesty to the Academic Judiciary. For more comprehensive
information on academic integrity, including categories of academic dishonesty, please
refer to the academic judiciary website,
http://www.stonybrook.edu/commcms/academic_integrity/index.html.
**Critical Incident Management Statement:
**

Stony Brook University expects students
to respect the rights, privileges, and property of other people. Faculty are required
to report to the Office of Judicial Affairs any disruptive behavior that interrupts
their ability to teach, compromises the safety of the learning environment, or
inhibits students' ability to learn.