MATH 525a SAMPLE FINAL EXAM Fall 2013 Prof. Alexander

MATH 525a
SAMPLE FINAL EXAM
Fall 2013
Prof. Alexander
NOTES:
(i) This exam is from several years ago. This semester problem (5) was given as homework,
and (4) is similar to Ch. 3 #12.
(ii) The final is in the lecture room, Monday December 16, 11 am–1 pm. It is open book,
meaning you can use the Folland text, your lecture notes, homework and solutions. You
cannot use other published materials (books, material from the internet, etc.) The final
covers the full semester, but with more emphasis on the second half (after the midterm.)
(iii) It’s not expected that one could do all 6 problems in 2 hours, but it should be feasible
to do at least 4.
(1) Let X be a set and A ⊂ X. Define the collection of sets which contain either “all or
none” of A:
M = {E ⊂ X : A ⊂ E or E ∩ A = φ}.
(a) Show that M is a σ-algebra.
(b) Which functions f : X → R are measurable on (X, M)? Give a description that is
specific to this problem, not just the general definition of measurable function.
(2) Let fn , f, g be real-valued measurable functions on (X, M, µ). Suppose fn → f in
measure and |fn | ≤ g a.e. for all n.
(a) Show that |f | ≤ g.R
R
(b) Assume also that g 2 dµ < ∞. Show that (fn − f )2 dµ → 0.
(3) Let F1 , F2 , ... and F all be in NBV, and suppose Fk → F pointwise. Show that
TF (x) ≤ lim inf TFk (x) for all x.
k→∞
(See page 102, 103 for the definitions of TF and NBV.) HINT: Show that for arbitrary > 0,
TF (x) − ≤ lim inf k TFk (x).
(4) Suppose µ1 , ν1 are positive finite measures on (X1 , M1 ) and µ2 , ν2 are positive finite
measures on (X2 , M2 ), with ν1 µ1 and ν2 µ2 . Show that ν1 × ν2 µ1 × µ2 . HINT:
It’s enough to show that for some f ,
Z
ν1 × ν2 (E) =
f d(µ1 × µ2 )
E
for all E ∈ M1 × M2 . See if you can guess what f should be, then prove you’re right.
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(5) Let us say that two signed measures ν1 , ν2 on (X, M) are compatible if there exists a
decomposition X = P ∪ N which is a Hahn decomposition for both ν1 and ν2 . According to
Proposition 3.14 page 94, for ν1 , ν2 finite signed measures,
(∗)
|ν1 + ν2 |(E) ≤ |ν1 |(E) + |ν2 |(E) for all E.
dν
Let µ = |ν1 | + |ν2 | and let fj = dµj , j = 1, 2. We write [fj > 0] as a shorthand for
{x ∈ X : fj (x) > 0}, and similarly for [fj < 0]. Show that the following are equivalent:
(i) ν1 and ν2 are compatible;
(ii) equality holds in (*) for all E;
(iii) µ([f1 > 0] ∩ [f2 < 0]) = µ([f1 < 0] ∩ [f2 > 0]) = 0.
HINT: Show (i) ⇔ (iii) and (ii) ⇔ (iii). To show (ii) implies (iii), show “not (iii)” implies
“not (ii).”
(6) Let (X, M, µ) be a measure space, with µ finite, and let ϕ be a bounded linear functional
on L1 (µ). Define ν on M by ν(E) = ϕ(χE ).
(a) Show that ν is a finite signed measure. HINT: Use Prop. 5.2.
(b) Show that ν µ.
R dν
dν
dν
is a bounded function. HINT: dµ
≤ M a.e. if and only if E dµ
dµ ≤
(c) Show that dµ
R
M dµ for all E. (Why? Explain, if you
this hint.)
E
R use
dν
(d) Show that ϕ is given by ϕ(f ) = f dµ dµ for all f ∈ L1 (µ). HINT: Consider f = χE
first.
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