Abbreviated summary of the second half of
221 Analysis 2, 2008-09
Disclaimer: This is intended to give you an overview of the second half of the course, and
many of the statements are rather vague, being intended to evoke the precise statements
given in the course. It is not an exhaustive list of examinable material. With a few
exceptions, noted below, anything we covered in the lectures and exercises, including
proofs, should be considered examinable. You will not be asked to prove results marked
no proof, but you are expected to be able to state and apply them.
Measure theory
extended real line, ring of subsets, measure, countable additivity
Measures:
• are increasing;
• respect set differences: m(B \ A) = m(B) − m(A) if m(B) < ∞ and
A ⊆ B;
• are countably subadditive (on any countable unions);
• respect countable increasing unions;
• respect countable decreasing intersections for sets of finite measure.
outer-measure of a measure
Suppose m is a measure on a ring of subsets of a set X. The
outer-measure m∗ of a measure m:
• is defined for all subsets of X;
• extends m;
• is countably subadditive;
• but it generally fails to be a measure.
m-measurable subset of X, σ-algebra
no proof Theorem (The Extension Theorem). Let m : A → [0, ∞] be a measure
where A is a ring of subsets of a set X, and let M denote the collection of
all m-measurable subsets of X.
(i). A ⊆ M,
(ii). M is a σ-algebra, and
(iii). m∗ |M : M → [0, ∞] is a measure which extends m.
σ-finite measure
Corollary. For any measure m : A → [0, ∞], there is an extension of m to
a measure on the σ-algebra M of all m-measurable sets. If m is σ-finite,
then this extension is unique.
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Lebesgue measure on R
interval ring, length measure
no proof Theorem. The length measure m : I → [0, ∞] is a (well-defined) σ-finite
measure.
Lebesgue measurable subset of R, Lebesgue measure
Notation for this section:
M = {Lebesgue measurable subsets of R}, m =Lebesgue measure on R.
• Lebesgue measure is translation invariant. . .
• . . . and respects dilations: m(cA) = |c|m(A) for A ∈ M and c ∈ R.
no proof
• A non-measurable subset of R can be found with the Axiom of Choice.
σ-algebra generated by a collection of subsets of R, Borel subsets of R
Theorem. The collection of Borel subsets of R is equal to σ(I) where I is
the interval ring.
Theorem. For any Lebesgue measurable set A ∈ M, there is a Borel set B
with A ⊆ B and m(B \ A) = 0.
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Integration with respect to a measure
measure space (X, M, m)
measurable set, measurable function, characteristic function, simple function
• the class of measurable functions is closed under sums and
multiplication by constants or by χA for A ∈ M.
the integral
•
R
R
ϕ dm of a simple nonnegative measurable function ϕ
χA dm = m(A) for A ∈ M.
• this integral respects sums and constant multiples
the integral
R
f dm of a nonnegative measurable function f
• this integral respects order: f ≤ g =⇒
R
f dm ≤
R
g dm
monotone increasing and decreasing sequences of functions, fn ↑ f , fn ↓ f
• the (pointwise) limit of a monotone sequence (fn ) always exists;
• if each fn is measurable then so is lim fn
n→∞
• Monotone Convergence Theorem: the integral respects limits of
monotone increasing sequences of nonnegative measurable functions.
• Every nonnegative measurable function is the limit of a monotone
increasing sequence of nonnegative simple measurable functions
(constructed by partitioning the domain of the function).
• So for non-negative measurable functions, the integral respects sums,
series and non-negative constant multiples.
R
• Also, λ(A) = A f dm defines a measure for a non-negative
measurable function f , which has pleasant consequences.
m-null, or null, f = g almost everywhere
If f, g, fn are non-negative measurable functions then
R
• f dm = 0 ⇐⇒ f = 0 a.e.;
R
• if A is m-null then A f dm = 0;
R
R
• if f = g a.e. then f dm = g dm.
R
R
• a.e. MCT: if fn ↑ f a.e. then fn dm → f dm as n → ∞.
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f + , f − , integrability of f : X → R with respect to m,
R
the integral f dm of an integrable function f
• If f is measurable, then f is integrable ⇐⇒
R
|f | dm < ∞.
• This integral respects sums and constant multiples of integrable
functions.
R
R
• If f = g a.e. then f dm = g dm.
R
R
• | f dm| ≤ |f | dm.
integrability and the integral of a complex-valued function
R
R
• If f : X → C is integrable then | f dm| ≤ |f | dm.
lim sup, lim inf (of sequences of functions and of extended real numbers).
Theorem (Fatou’s lemma). If (fn )n≥1 is a sequence of nonnegative
measurable functions fn : X → [0, ∞] then
Z
Z
lim inf fn dm ≤ lim inf fn dm.
n→∞
n→∞
Theorem (The Dominated Convergence Theorem). Suppose that (fn )n≥1 is
a sequence of integrable functions fn : X → R and that fn → f as n → ∞.
If there is an integrable function g : X → R with |fn | ≤ g for all n ≥ 1, then
Z
Z
f is integrable, and
f dm = lim
fn dm.
n→∞
Consequently:
• under a suitable hypothesis, the integral respects summation of series;
• under suitable hypotheses (on integrability, differentiability and the
existence of a dominating function for the derivative), we can
“differentiate under the integral sign”.
Lebesgue integration on R
the Lebesgue integral
• The Fundamental Theorem of Calculus holds for the Lebesgue
integral;
• and given suitable differentiability hypotheses, we can compute
Lebesgue integrals by substitution.
• The Lebesgue integral is translation invariant and respects dilations.
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Multiple integration
Let (X, L, `) and (Y, M, m) be measure spaces.
rect(L, M), Arect (L, M)
• Arect (L, M) is a ring of subsets of X × Y
• we define π : Arect (L, M) → [0, ∞] by π(A × B) = `(A)m(B) and
extending to disjoint unions to (try to) make it countably additive.
This works, and π is a well-defined measure.
product σ-algebra L × M, product measure ` × m (which is a measure)
Monotone classes
monotone class, mon(S)
no proof
• Monotone class lemma: If A is a ring of subsets of X then
mon(A) is also a ring of subsets of X.
• So any ring of subsets of X containing X has σ(A) = mon(A).
• So L × M = mon(Arect (L, M)).
Integration using product measure
Ex and E y for E ⊆ X × Y and x ∈ X, y ∈ Y
• If E ∈ L × M then Ex ∈ M for every x ∈ X, and E y ∈ L for every
y ∈Y.
Now suppose that l and m are σ-finite.
R
R
no proof
• l × m(E) = X m(Ex ) dl(x) = Y l(E y ) dm(y) for a measurable set E.
R R
R
• Tonelli’s
theorem:
F
(x,
y)
dm(y)dl(x)
=
F d(l × m) =
X
Y
X×Y
R R
F (x, y) dl(y) dm(x) for F nonnegative and measurable.
Y X
• Fubini’s theorem: the same formula is true (suitably interpreted) if
F is real-valued and integrable.
Lebesgue measure and integration on Rn
Lebesgue measure on Rn , Lebesgue measurable subsets of Rn ,
R
Rn
f (x) dx
• If
and f : Rn → R is integrable then
R A ∈ Mn (R) is invertible
R
1
f (Ax) dx = | det A| Rn f (x) dx.
Rn
• So if µ is Lebesgue measure on Rn then µ(A(S)) = | det A|µ(S) for
any Lebesgue measurable S ⊆ Rn and any A ∈ Mn (R).
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