Real Analysis Comprehensive Examination–Math 921/922
Thursday, January 21, 2016, 12:30-6:30p.m., Avery Hall 347
• Work 6 out of 8 problems. • Each problem is worth 20 points. • Write on one side of the paper only and hand your work in order.
• Throughout the exam, the Lebesgue measure is denoted by m, BX denotes the Borel σ-algebra on a metric space X, and
(X, M, µ) denotes a general measure space.
(1) Let X be an uncountable set. For each E ∈ P(X) define µ∗ (E) = 0 if E is countable and
µ∗ (E) = 1 if E is uncountable. Let A = {E ⊂ X : E is countable or E c is countable}.
a) Show that A is σ-algebra.
b) Prove that µ∗ is an outer measure on X, and determine the class of µ∗ -measurable sets.
(2) Let E ⊂ R be a countable set and f : R → R be a continuous function; except for
discontinuities at each point of E. Assume that some γ ∈ R is not in the range of f . Prove
that f is Borel measurable. (Hint: first, prove the Borel-measurability of a modified function
f˜, that agrees with f on E c and maps every point of E to the value γ.)
(3) Let f : X → R be such that f ∈ L1 (X,
that: for every > 0 there exists a
R M, µ). Prove
R
set E ∈ M with µ(E) < ∞, such that E |f |dµ > X |f |dµ − .
x
n
(4) Let fn (x) =
sin
, where 0 < x < ∞ and n ∈ N. Show all technical details in
x(1 + xR2 )
n
evaluating: limn→∞ (0,∞) fn dm.
(5) Let ν be Rthe signed measure
on B(0,∞) that is given by:
P
ν(E) := E e−x dm + {n∈N: n∈E} (−1)n 21n , E ∈ B(0,∞) .
With justification, find:
a) ν(0, 1].
b) The Hahn decomposition for ν.
c) The Jordan decomposition of ν.
d) |ν|, the total variation of ν.
e) The Lebesgue decomposition of |ν| with respect to m.
f) limr→0+ ν(Er )/m(Er ), where {Er } is a family of sets that shrinks nicely to x ∈ (0, ∞).
R
(6) Let g : (0, ∞) → R be a bounded Borel measurable function with (0,2) g dm = −π. Let
f (x, y) = χE (x, y) x−1 g 2y
, where E = {(x, y) ∈ (0, 1)2 : y ∈ (0, x), x ∈ (0, 1)}. Carefully
x
explain why f is Borel measurable on (0, 1) × (0, 1), and use Fubini-Tonelli’s Theorems to
evaluate (with all justifications) the single integral:
Z
f (x, y) d(m × m).
(0,1)×(0,1)
(7) Let fn , f ∈ X := Lp ([0, 1], B[0,1] , m) for some 1 < p < ∞.
a) Prove that if fn → f in X then fn converges to f weakly in X.
R
R
b) Prove that if the sequence (fn ) is bounded in X and limn→∞ E fn dµ = E f dµ for every
E ∈ B[0,1] , then fn converges weakly to f in X.
(8) a) LetR f : R → R be a non-decreasing function. Prove that: whenever −∞ < a < b < ∞,
then [a,b] f 0 (t) dm(t) ≤ f (b) − f (a).
b) Let f : R → RR be a non-decreasing function and such that: f (−∞) = −1 and f (+∞) = 1.
Prove that: If R f 0 (t) dm(t) = 2, then f is absolutely continuous on every closed bounded
segment [a, b]. (Hint: Use part a)).