Exam1

Michael Johnson Dept of Math and Comp Sci

Math 510 First Midterm Exam

Date: Nov. 24, 2008 Duration: 75 minutes

1. (7 pts) Define each of the following: (a) m∗ (A). (b) A is measurable. (c) φR is a simple function. (d) φ (when φ is a simple function which vanishes outside a set of finite measure). (e) f is a measurable function. Z (f) L g (when g is a bounded function and m(E) < ∞). Z E (g) f (when f is a bounded measurable function and m(E) < ∞). E

2. (7 pts) Let C denote the collection of singleton’s {q}, where q is rational; that is, C = {{q} : q ∈ Q}. Show that the smallest σ-algebra containing C is A = {A : A ⊂ Q or A ⊂ R ∼ Q}. 3. (6 pts) Show that the function f (x) =



sin x

if x ∈ Q

2 + cos x

if x 6∈ Q

is measurable. 4. (7 pts) Let f and g be nonnegative, bounded measurable functions defiend on E = [−1, 1], and suppose that φ2 ≥ g on E whenever φ is a simple function satisfying φ ≥ f on E. For each statement, give Z Z either a proof or a counter-example. 2 (a) f ≥ g ZE ZE (b) g2 ≥ f ZE ZE (c) f2 ≤ g ZE ZE (d) g2 ≤ f E

E

5. (7 pts) Let {fn } be a sequence of measurable functions defined on R such that for all m, n ∈ N, fn = fm a.e. on R. Prove that there exists a measurable set A, with m(A) = 0, such that fn (x) = fm (x) for all x 6∈ A and m, n ∈ N. 6. (6 pts) Let {fn } be a sequence of bounded measurable functions, defined on E = [−1, 1], which 1 for all x ∈ E. converge to f (x) = 1 + x2 Z (a) Show that

f = π/2. Z (b) Prove that if lim fn = 2, then sup fn is an unbounded function. E

n→∞

E

n