Download 1. (20%)State the following theorems without proof. (a) Lebesgue`s

1. (20%)State the following theorems without proof.
(a) Lebesgue’s Dominated Convergence Theorem.
(b) Riesz Representation Theorem for Positive Linear Functionals.
(c) The Theorem of Lebesgue-Radon-Nikodym.
(d) Fubini Theorem.
2. (10%)Let X be a topological space and f : X → [−∞, +∞]. Then f is upper (resp. lower) semicontinuous
if {x ∈ X : f (x) < α} (resp. {x ∈ X : f (x) > α} ) is open for every real α.
(a) Prove that if f1 and f2 are upper semicontinuous on X then f1 + f2 is upper semicontinuous on X.
(b) Prove that f is continuous on X if and only if f is both upper and lower semicontinuous on X.
3. (10%)Let
m be a σ-algebra in X, Y
be a topological space and f : X → Y .
m} is a σ-algebra in Y .
(b) Prove that if f is measurable then f −1 (E) ∈ m for all Borel set E in Y .
4. (10%)Let (X, m, µ) be a measure space, f : X → [0, +∞] be a measurable function and
Z
f dµ (E ∈ m).
ϕ(E) =
(a) Prove that Ω = {E ⊂ Y : f −1 (E) ∈
E
(a) Prove that ϕ is a positive measure on
m.
(b) Prove that
Z
gdϕ =
X
Z
gf dµ
X
for every measurable g on X with range in [0, +∞].
5. (10%)
(a) Prove that if µ is a finite positive measure and 1 < p < q < +∞ then Lq (µ) ⊂ Lp (µ).
(b) Prove that if 1 < p < q < +∞ then l p ⊂ lq .
6. (10%)Let (X,
m, µ) be a measure space, w : X → (0, +∞] be a measurable function and
Z
wdµ (E ∈ m).
µ
e(E) =
E
(a) Prove that
i. µ(E) = 0 if and only if µ
e(E) = 0.
ii. µ is concentrated on A if and only if µ
e is concentrated on A.
(b) Prove that if λ is a measure on
m then
i. λ⊥µ if and only if λ⊥e
µ.
ii. λ µ if and only if λ µ
e.
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7. (30%)Let f (x) = xe−x (−∞ < x < +∞) and µ(E) =
subset of R1 and m is the Lebesgue measure in R1 .
R
E
f dm where E is any Lebesgue measurable
(a) Find µ(R1 ) and the total variation |µ|(R1 ) of µ.
(b) Find µ+ and µ− . (µ = µ+ − µ− is the Jordan decomposition of µ.)
(c) Find a Hahn decomposition (A, B) of R1 induced by µ.
(d) Find h such that dµ = hd|µ|, the polar representation of µ.
(e) Find the Lebesgue decomposition (µa , µs ) of µ relative to m.
(f) Find the Radon-Nikodym derivative dµa /dm of µa with respect to m.
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