MATH 507/420 - Assignment #4
Due on Friday November 4, 2011
Name —————————————–
Student number —————————
1
Problem 1:
Let (X, M, µ) be a measure space and g a nonnegative function that
is integrable over X. Define
Z
g dµ for all E ∈ M.
ν(E) =
E
Let f be a measurable nonnegative function on X. Show that
Z
Z
f dν =
f g dµ.
X
X
2
Problem 2:
Let (X, M, µ) be a measure space and f a bounded function on X that
vanishes outside a set of finite
measure. ZAssume
Z
sup
ψ dµ = inf
X
ϕ dµ,
X
where ψ ranges over all simple functions on X for which ψ ≤ f on X
and ϕ ranges over all simple functions on X for which f ≤ ϕ on X.
Prove that f is measurable with respect to completion of (X, M, µ).
3
Problem 3:
Let S be an algebra of subsets
a set X. We say that a function
Pof
n
φ : X → R is S-simple if φ = k=1 ak χAk , where each Ak ∈ S. Let µ
be a premeasure on S and µ its Carathéodory extension. Given > 0
and a function f that is integrable over X with respect to µ, show that
there is an S-simple function φ such that
Z
|f − φ| d µ < .
X
4
Problem 4:
Let (X, M, µ) be a complete measure space and {fn } a sequence of
measurable functions on X which converges pointwise a.e. to a function
f and {gn } a sequence of integrable functions over X which converges
pointwise
a.e. toR an integrable function g. Assume that |fnR| ≤ gn and
R
lim
g
R n X n dµ = X g dµ. Show that f is integrable and limn X fn dµ =
f dµ.
X
5
Problem 5:
A sequence {fn } of real valued measurable functions on X is said to
converge in measure to a measurable function f provided that for each
r > 0,
lim µ({x ∈ X s.t. |fn (x) − f (x)| > r}) = 0.
n→∞
(a) Deduce the Lebesgue Dominated Convergence Theorem from the
Vitali Convergence Theorem .
(b) Show that almost everywhere pointwise convergence can be replaced by convergence in measure in the Lebesgue Dominated Convergence Theorem and the Vitali Convergence Theorem .