THEOREMS, ETC., FOR MATH 516
When a statement in this list corresponds to a result in the book, I
have listed the statement from the book in parentheses. Note that, for
example, Proposition 1.2 means Proposition 2 from Chapter 1.
Theorem 1 (=Corollary 7.41, Arzela-Ascoli Theorem). Let (X, ρ) be
a separable metric space, and let F be an equicontinuous, uniformly
bounded family of continuous, real-valued functions defined on X. Then
every sequence in F has a subsequence that converges at each point in
X. The limit function is continuous, and the subsequence converges
uniformly (to the limit function) on any compact subset of X.
Proposition 2 (=Proposition 8.2). Let E, A, and B be subsets of a
topological space (X, T ). Then
E ⊂ Ē, Ē¯ = Ē, A ∪ B = Ā ∪ B̄.
Also, E is closed if and only if E = Ē, and Ē is the set of all points
of closure of E.
Further
E o ⊂ E,
(E o )o = E o ,
(A ∩ B)o = Ao ∩ B o ,
E is open if and only if E = E o , and (E o )o = E 0 .
Finally, ({E)o = {(Ē).
Proposition 3 (=Proposition 8.3). Let (X, T ) be a topological space
and let A1 and A2 be subsets of X with X = A1 ∪ A2 . Finally, let
(Y, S ) be a topological space and let f : X → Y be a function such that
f|A1 and f|A2 are continuous. If A1 and A2 are both open or both closed,
then f is continuous.
Proposition 4 (=Proposition 8.4). If C is a collection of subsets of a
non-empty set X, then there is a weakest topology T such that C ⊂ T .
Proposition 5 (=Proposition 8.5). Let B be a collection of subsets of
a non-empty set X. Then B is a base for a topology T on X if and
only if the following two conditions are satisfied.
(i) For every x ∈ X, there is a B ∈ B such that x ∈ B.
(ii) For every B1 and B2 in B and every x ∈ B1 ∩ B2 , there is a
B ∈ B such that x ∈ B ⊂ B1 ∩ B2 .
Date: May 1, 2009.
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THEOREMS, ETC., FOR MATH 516
In addition, T is the weakest topology such that B ⊂ T .
Proposition 6 (=Proposition 8.6). A topological space is T1 if and
only if every one-point set is closed.
Lemma 7 (=Lemma 8.7, Urysohn’s Lemma). Let (X, T ) be a normal
topological space and let A and B be disjoint, closed, non-empty subsets
of X. Then there is a continuous function f : X → [0, 1] such that
f (A) = {0} and f (B) = {1}.
Theorem 8 (=Theorem 8.8, Tietze Extension Theorem). Let (X, T )
be a normal topological space, let A be a closed subset of X, and let
f be a continuous real-valued function defined on A. Then there is a
continuous real-valued function g defined on X such that g|A = f .
Proposition 9 (=Proposition 8.10). Let (X, T ) and (Y, S ) be topological spaces, and let f : X → Y be continuous. If X is connected,
then f (X) is connected.
Proposition 10 (=Proposition 8.11). Let (X, T ) be a connected topological space and let f be a continuous real-valued function on X. If x
and y are points of X such that f (x) < f (y) and if c ∈ (f (x), f (y)),
then there is a point z ∈ X such that f (z) = c.
Proposition 11 (=Proposition 8.12). A non-empty subset of R is connected if and only if it is either an interval or a one-point set.
Proposition 12 (=Proposition 9.1). A topological space is compact if
and only if every collection of closed sets with the finite intersection
property has non-empty intersection.
Proposition 13 (=Proposition 9.2). A closed subset of a compact set
is compact. A compact subset of a Hausdorff space is closed.
Proposition 14 (=Proposition 9.4). The continuous image of a compact set is compact.
Proposition 15 (=Proposition 9.5). A one-to-one onto, continuous
map from a compact space to a Hausdorff space is a homeomorphism.
Proposition 16 (=Proposition 9.6). The continuous image of a countably compact set is countably compact. The continuous image of a Lindelöf space is Lindelöf.
Proposition 17 (=Proposition 9.7). A topological space is countably
compact if and only if it has the Bolzano-Weierstrass property.
Proposition 18 (=Proposition 9.8). A sequentially compact space is
countably compact. A first countable, countably compact space is sequentially compact.
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3
Proposition 19 (=Proposition 9.10). Let f be a real-valued, upper
semicontinuous function defined on a countably compact space. Then
f is bounded above and f attains its maximum value.
Proposition 20 (=Proposition 9.11, Dini). Let (X, T ) be a countably
compact space and let (fn ) be a decreasing sequence of real-valued, upper
semicontinuous functions defined on X. If lim fn (x) = 0 for all x ∈ X,
then (fn ) converges uniformly to zero.
Lemma 21 (=Lemma 9.12). Let X be a set and let A be a collection
of subsets of X with the finite intersection property. Then there exists
a maximal collection B of subsets of X with the finite intersection
property such that A ⊂ B.
Lemma 22 (=Lemma 9.13). Let X be a set and let B be a maximal
collection of subsets of X with the finite intersection property. Then
any intersection of finitely many elements of B is in B, and any subset
of B which has non-empty intersection with any element of B is an
element of B.
Theorem 23 (=Theorem 9.14, Tychonoff’s Theorem). The product of
a family of compact topological spaces is also compact (in the product
topology).
Proposition 24 (=Proposition 9.15). Let (X, T ) be a locally compact
Hausdorff space, and let K be a compact subset of X. Then there is an
open set O such that O is compact and K ⊂ O. For any such set O,
there is a continuous function f : X → [0, 1] such that f is zero in the
complement of O and f is one on K.
Proposition 25 (=Proposition 9.16). Let (X, T ) be a locally compact
Hausdorff space, let K be a compact subset of X, and let (Oλ )λ∈L be
an open cover of K. Then there is a partition of unity subordinate to
(Oλ )λ∈L consisting of finitely many continuous functions, each of which
has compact support.
Proposition 26 (=Proposition 9.29). Let (X, T ) be a compact topological space. If L ⊂ C(X) is a lattice such that h, defined by
h(x) = inf{f (x) : f ∈ L},
is continuous, then, for any ε > 0, there is g ∈ L such that 0 ≤ g−h < ε
in X.
Lemma 27 (=Lemma 9.31). Let L be a family of real-valued functions
defined on a set X. Suppose L separates points and that, for any f ∈ L
and any c ∈ R, c + f and cf are in L. Then, for any x 6= y in X and
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THEOREMS, ETC., FOR MATH 516
any real numbers a and b, there is f ∈ L such that f (x) = a and
f (y) = b.
Lemma 28 (=Lemma 9.32). Let (X, T ) be a compact topological space
and let L ⊂ C(X) be a lattice. Suppose L separates points and that,
for any f ∈ L and any c ∈ R, c + f and cf are in L. Then, for any
real numbers a ≤ b, any closed subset F of X, and any p ∈ X \ F ,
there is f ∈ L such that f ≥ a on X, f (p) = a, and f > b on F .
Proposition 29 (=Proposition 9.30). Let (X, T ) be a compact topological space and let L ⊂ C(X) be a lattice. Suppose L separates points
and that, for any f ∈ L and any c ∈ R, c + f and cf are in L. Then,
for any h ∈ C(X) and any ε > 0, there is g ∈ L such that 0 ≤ g −h < ε
on X.
Lemma 30 (=Lemma 9.33). For any ε > 0, there is a polynomial Pε
such that |P (s) − |s|| < ε for all s ∈ [−1, 1].
Theorem 31 (=Theorem 9.34, Stone-Weierstrass Theorem). Let (X, T )
be a compact space and let A ⊂ C(X) be an algebra that separates points
and contains the constant functions. Then A is dense in C(X).
Corollary 32 (=Corollary 9.35, Weierstrass Approximation Theorem). Every continuous function on a closed bounded subset of Rn can
be uniformly approximated by a polynomial.
Proposition 33 (=Proposition 10.2). A bounded linear operator is
uniformly continuous. If a linear functional is continuous at a point,
then it is bounded.
Proposition 34 (=Proposition 10.3). The set of all bounded linear
operators from a normed linear space to a Banach space is a Banach
space.
Theorem 35 (=Theorem 10.4, Hahn-Banach Theorem). Let X be a
vector space and let p be a real-valued function on X such that p(x+y) ≤
p(x) + p(y) for all x and y in X and p(αx) = αp(x) for all α ≥ 0 and
all x ∈ X. Suppose f is a linear functional defined on some subspace S
of X so that f (s) ≤ p(s) for all s ∈ S. Then there is a linear functional
F defined on X such that F (x) ≤ p(x) for all x ∈ X and F|S = f .
Proposition 36 (=Proposition 10.5). Let X be a vector space and let
p be a real-valued function on X such that p(x + y) ≤ p(x) + p(y) for
all x and y in X and p(αx) = αp(x) for all α ≥ 0 and all x ∈ X, and
let f be a linear functional defined on some subspace S of X. Suppose
G is a collection of linear operators on X such that for all A and B
in G, we have AB ∈ G and AB = BA. Suppose also that, for all
THEOREMS, ETC., FOR MATH 516
5
A ∈ G, all s ∈ S, and all x ∈ X, we have p(Ax) ≤ p(x), As ∈ S, and
f (As) = f (s). Then there is an extension F of f to a linear functional
defined on all of X with F (x) ≤ p(x) and F (Ax) = F (x) for all x ∈ X
and all A ∈ G.
Proposition 37 (=Proposition 10.6). Let X be a normed vector space
and let x ∈ X. Then there is a bounded linear functional F on X such
that F (x) = kF kkxk.
Proposition 38 (=Proposition 10.7). Let T be a subspace of a normed
linear space X, and suppose there are x ∈ X and δ > 0 such that
kx − tk > δ for all t ∈ T . Then there is a bounded linear functional F
on X such that kF k ≤ 1, F (x) = δ, and F (t) = 0 for all t ∈ T .
Theorem 39 (=Lemma 10.9+Proposition 10.10, Open Mapping Theorem). Let A be a bounded linear operator from a Banach space X onto
a Banach space Y . Then A is an open mapping.
Corollary 40 (=Proposition 10.11). Let X be a vector space, and
suppose k · k1 and k · k2 are two norms that make X into a Banach
space. If there is a constant c1 such that
kxk1 ≤ c1 kxk2
for all x ∈ X, then there is a constant c2 such that
kxk2 ≤ c2 kxk1
for all x ∈ X.
Theorem 41 (=Theorem 10.12, Closed Graph Theorem). Let A be a
linear operator from a Banach space X to a Banach space Y . Suppose
that whenever (xn ) is a convergent sequence in X (with limit x) such
that (Axn ) is a convergent sequence in Y (with limit y), we have y =
Ax. Then A is a bounded linear operator.
Proposition 42 (=Proposition 10.13, Uniform Boundedness Principle). Let F be a family of linear operators from a Banach space X to
a normed vector space Y . If, for every x ∈ X, there is a nonnegative
number Mx such that kf (x)kY ≤ Mx for all f ∈ F , then there is a
nonnegative constant M such that kf k ≤ M for all f ∈ F .
Proposition 43 (=Proposition 10.14). Let X be a vector space.If B)
is a family of subsets of X all containing 0 such that
(i) for all U and V in B0 , there is W ∈ B0 such that W ⊂ U ∩ V ,
(ii) for all U ∈ B0 and all x ∈ U , there is V ∈ B0 such that
x + V ⊂ U,
(iii) for all U ∈ B0 , there is V ∈ B0 such that V + V ⊂ U ,
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THEOREMS, ETC., FOR MATH 516
(iv) for all U ∈ B0 and all x ∈ X, there is a real number α such
that x ∈ αU ,
(v) for all U ∈ B0 and all α ∈ R such that 0 < |α| ≤ 1, we have
αU ⊂ U and αU ∈ B0 ,
then B0 is a base at 0 for a topology that makes X into a topological
vector space. Conversely, if T is a topology that makes X into a topological vector space, then there is a base at 0, B0 , satisfying conditions
(i)–(v). In addition (X, T ) is Hausdorff if and only if
(vi) the only element in U for all U ∈ B0 is 0.
Proposition 44 (=Proposition 10.15, Tychonoff). Let X be a Hausdorff topological space of finite dimension n. Then X is topologically
isomorphic to Rn .
Proposition 45 (=Proposition 10.16). A subspace of a Banach space
is weakly closed if and only if it is strongly closed.
Theorem 46 (=Theorem 10.17, Alagoulu’s Theorem). The closed unit
ball in a Banach space is weakly compact.
Lemma 47 (=Lemma 10.18). If K1 and K2 are convex subsets of a
vector space X and if λ ∈ R, then K1 ∩ K2 , λK1 , and K1 + K2 are also
convex.
Lemma 48 (=Lemma 10.19). Let K be a convex subset of a vector
space X and suppose that 0 is an internal point of K. Then
(1) p(λx) = λp(x) for all λ ≥ 0 and all x ∈ X.
(2) p(x + y) ≤ p(x) + p(y) for all x and y in X.
(3) If p(x) < 1, then x ∈ K. If x ∈ K, then p(x) ≤ 1.
Theorem 49 (=Theorem 10.20). Let K1 and K2 be two disjoint convex
subsets of a vector space X. If one has an internal point, then the two
sets are separated by a non-zero linear functional.
Proposition 50 (=Proposition 10.22). Let X be a locally convex topological vector space, and let F be a closed convex subset of X. Then,
for any x0 ∈ X \ F , there is a continuous linear functional f on X
such that
f (x0 ) < inf{f (x) : x ∈ F }.
Lemma 51 (=Lemma 10.25). Let f be a continuous linear functional
on a closed convex subset K of a topological vector space. Then the set
S = {x : f (x) = max f }
is a supporting set for K.
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Theorem 52 (=Theorem 10.26, Krein-Milman). Let K be a nonempty compact convex subset of a locally convex topological vector space.
Then K is the closed convex hull of its extreme points.
Proposition 53 (=Proposition 10.27). Every Hilbert space H has a
complete orthonormal system. If H is separable, then every orthonormal system is countable. Moreover, if H is separable and (ϕn ) is a
complete orthonormal system for K, then, for every x ∈ H, we have
x=
∞
X
(x, ϕn )ϕn
n=1
and
2
kxk =
∞
X
(x, ϕn )2 .
n=1
Theorem 54 (Projection Theorem, not in Royden). Let M be a closed
subspace of a Hilbert space X. Then for every x ∈ X, there is a unique
p(x) ∈ M such that x − p(x) is orthogonal to every element of M .
Theorem 55 (=Proposition 10.28, Riesz Representation Theorem).
If f is a bounded linear functional on a Hilbert space X, then there
is a unique z ∈ X such that f (x) = (x, z) for all x ∈ X. Moreover
kzk = kf k.
Proposition 56 (=Proposition 11.1). Let (X, B, µ) be a measure space.
If A and B are measurable with A ⊂ B, then µA ≤ µB.
Proposition 57 (=Proposition 11.2). Let (X, B, µ) be a measure space
and let (Ei ) be a sequence of measurable sets with Ei+1 ⊂ Ei for all i
and µE1 < ∞. Then
̰ !
\
µ
Ei = lim µEn .
i=1
Proposition 58 (=Proposition 11.3). Let (X, B, µ) be a measure space
and let (Ei ) be a sequence of measurable sets. Then
̰ !
∞
X
[
µEi .
µ
Ei ≤
i=1
i=1
Proposition 59 (=Proposition 11.4). Let (X, B, µ) be a measure space.
Then there is a unique complete measure space (X, B0 , µ0 ) such that
(1) B ⊂ B0 ,
(2) µE = µ0 E for all E ∈ B,
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THEOREMS, ETC., FOR MATH 516
(3) E ∈ B0 if and only if there are sets B and C in B and a subset
A of C such that µC = 0 and E = A ∪ B.
Proposition 60 (=Proposition 11.5). Let (X, B) be a measurable
space, and let f be an extended real-valued function on X. Then the
following conditions are equivalent.
(1) {x : f (x) < α} ∈ B for all αinR.
(2) {x : f (x) > α} ∈ B for all αinR.
(3) {x : f (x) ≤ α} ∈ B for all αinR.
(4) {x : f (x) ≥ α} ∈ B for all αinR.
Theorem 61 (=Theorem 11.6). Let (X, B) be a measurable space. If
f and g are measurable functions and if c ∈ R, then
f + c,
cf,
f + g,
f g,
min{f, g},
max{f, g}
are measurable. If (fn ) is a sequence of measurable functions, then
sup fn ,
inf fn ,
lim fn ,
lim fn
are measurable.
Proposition 62 (=Proposition 11.7). Let (X, B, µ) be a measure space,
and let f be a nonnegative measurable function. Then there is an increasing sequence (ϕn ) of simple functions such that ϕn → f . If X is
σ-finite, then each ϕn can be chosen to vanish outside a set of finite
measure.
Proposition 63 (=Proposition 11.8). Let (X, B, µ) be a complete
measure space, and let f be a measurable function. If g is an extended
real-valued function with f = g a.e., then g is measurable.
Lemma 64 (=Lemma 11.9). Let (X, B) be a measurable space, let D
be a dense subset of R, and suppose that, for each α ∈ D, there is a
measurable set Bα such that α < β implies that Bα ⊂ Bβ . Then there
is a unique measurable function f such that f (x) ≤ α for all x ∈ Bα
and f (x) ≥ α for all x ∈ X \ Bα .
Proposition 65 (=Proposition 11.10). Let (X, B, µ) be a measure
space, let D be a dense subset of R, and suppose that, for each α ∈ D,
there is a measurable set Bα such that α < β implies that µ(Bα \ Bβ ) =
0. Then there is a measurable function f such that f ≥ α a.e. on Bα
and f ≤ α a.e. on X \ Bα . If g is any other function with this property,
then f = g a.e.
Lemma 66 (=Theorem 11.11 , Fatou’s Lemma). Let (fn ) be a sequence
of nonnegative measurable functions and suppose that f = lim fn exists
THEOREMS, ETC., FOR MATH 516
a.e. Then
Z
9
Z
f dµ ≤ lim
E
fn dµ
E
for any measurable set E.
Theorem 67 (=Theorem 11.12, Monotone Convergence Theorem).
Let (fn ) be an increasing sequence of nonnegative measurable functions,
and set f = fn . Then
Z
Z
f dµ = lim fn dµ
E
E
for any measurable set E.
Proposition 68 (=Proposition 11.13). Let f and g be nonnegative
measurable functions, and let a and b be nonnegative constants. Then
Z
Z
Z
(af + bg) dµ = a f dµ + b g dµ
E
E
E
R
R
for any measurable E. In addition, E f ≥ 0 and E f = 0 if and only
if f = 0 a.e. on E.
Corollary 69 (=Corollary 11.14). Let (fn ) be a sequence of nonnegative measurable functions. Then
Z X
∞
∞ Z
X
fn dµ =
fn dµ
E n=1
n=1
E
for any measurable E.
Proposition 70 (=Proposition 11.15). Suppose f and g are integrable
functions and E is a measurable set.
(1) If a and b are real numbers, then
Z
Z
Z
(af + bg) dµ = a f dµ + b g dµ.
E
E
E
(2) If h is a measurable function with |h| ≤ |f | a.e., then h is
integrable.
R
R
(3) If f ≥ g a.e., then E f dµ ≥ E g dµ.
Theorem 71 (=Theorem 11.16, Lebesgue Dominated Convergence
Theorem). Let (fn ) be a sequence of measurable functions with f =
lim fn a.e., and suppose that there is a nonnegative integrable function
g such that |fn | ≤ g a.e. for all n. Then f is integrable and
Z
Z
f dµ = lim fn dµ.
E
E
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Proposition 72 (=Proposition 11.17). Let (µn ) be a sequence of measures on a measurable space (X, B) which converges setwise to a measure µ. Then µ is a measure on (X, B). In addition, if (fn ) is a
sequence of nonnegative measurable functions such that f = lim fn exists, then
Z
Z
f dµ ≤ lim
fn dµn .
E
E
Proposition 73 (=Proposition 11.18). Let (µn ) be a sequence of measures on a measurable space (X, B) which converges setwise to a measure µ. Suppose that (fn ) and (gn ) are sequences of measurable functions such that f = lim fn and g = lim gn exist everywhere. Suppose
also that |fn | ≤ gn for all n and that
Z
Z
g dµ = lim gn dµn < ∞.
Then
Z
Z
f dµ = lim
fn dµn .
Lemma 74 (=Lemma 11.19). Every measurable subset of a positive
set is positive. The union of countably many positive sets is positive.
Lemma 75 (=Lemma 11.20). Let (X, B, ν) be a signed measure space
and let E be a measurable set with 0 < νE < ∞. Then there is a
positive set A with A ⊂ E and νA > 0.
Theorem 76 (=Proposition 11.21, Hahn Decomposition Theorem).
Let (X, B, ν) be a signed measure space. Then there are disjoint measurable sets A and B with A ∪ B = X such that A is a positive set and
B is a negative set.
Proposition 77 (=Proposition 11.22). Let (X, B, ν) be a signed measure space. Then there are two mutually singular measures ν + and ν −
such that ν = ν + − ν − . Moreover, there is only pair of such measures.
Theorem 78 (=Theorem 11.23, Radon-Nikodym Theorem). Let (X, B)
be a measurable space and let µ and ν be two measures on B. If µ is
σ-finite and if ν is absolutely continuous with respect to µ, then there
is a nonnegative measurable function f such that
Z
νE =
f dµ
E
for all measurable sets E. If g is any other such function, then f = g
a.e.[µ].
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11
Theorem 79 (=Theorem 11.24, Lebesgue Decomposition). Let (X, B, µ)
be a σ-finite measure space, and let ν be a σ-finite measure on (X, B).
Then there are unique measures ν0 and ν1 such that ν0 ⊥ µ, ν1 ¿ µ
and ν = ν0 + ν1 .
Theorem 80 (=Theorem 11.25). For 1 ≤ p ≤ ∞, Lp (µ) is a Banach
space. For p1 + 1q = 1, if f ∈ Lp (µ) and g ∈ Lq (µ), then f g ∈ L1 (µ)
and
¯ Z
¯Z
¯
¯
¯ f g dµ¯ ≤ |f g| dµ ≤ kf kp kgkq .
¯
¯
Proposition 81 (=Proposition 11.26). Let 1 ≤ p < ∞. For every
f ∈ Lp (µ) and every ε > 0, there is a simple function ϕ which vanishes
outside a set of finite measure such that
kf − ϕkp < ε.
Lemma 82 (=Lemma 11.27). Let (X, B, µ) be a finite measure space
and let 1 ≤ p < ∞. If g ∈ L1 (µ) and if there is a constant M such that
¯Z
¯
¯
¯
¯ gϕ dµ¯ ≤ M kϕkp
¯
¯
for all simple functions ϕ, then g ∈ Lq (µ) for p1 + 1q = 1, and kgkq ≤ M .
Lemma 83 (=Lemma 11.28). Let (X, B, µ) be a measure space, let
(En ) be a sequence of disjoint measurable sets, let (fn ) be a sequence
in LpP
(µ) for some p ∈ [1, ∞) such thatP
fn = 0 a.e outside En , and set
f = Pfn . Then f ∈ Lp if and only if
kfn kp P
< ∞. In this case, the
sum
fn converges in Lp (µ) to f and kf kpp =
kfn kpp .
Theorem 84 (=Theorem 11.29, Riesz Representation). Let 1 ≤ p <
∞ and let (X, B, µ) be a σ-finite measure space. Then, for every
bounded linear function F on Lp (µ), there is a unique g ∈ Lq (µ) (with
1
+ 1q = 1) such that
p
Z
F (f ) = f g dµ
for all f ∈ Lp (µ). In addition, kgkq = kF k.
Theorem 85 (=Theorem 11.30, Riesz Representation). Let 1 < p <
∞ and let (X, B, µ) be a measure space. Then, for every bounded linear
function F on Lp (µ), there is a unique g ∈ Lq (µ) (with p1 + 1q = 1) such
that
Z
F (f ) = f g dµ
for all f ∈ Lp (µ). In addition, kgkq = kF k.
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THEOREMS, ETC., FOR MATH 516
Theorem 86 (=Theorem 12.1). Let µ∗ be an outer measure for a set
X. Then B, the collection of all measurable sets, is a σ-algebra, and
µ̄, the restriction of µ∗ to B, is a complete measure on B.
Lemma 87 (= Lemma 12.2). Let µ be a measure on A , an algebra
of subsets of a setSX, and let (Ai ) P
be a sequence of elements of A . If
A ∈ A and A ⊂ Ai , then µA ≤
µAi .
Corollary 88 (=Corollary 12.3). The restriction of µ∗ to A is just µ.
Lemma 89 (= Lemma 12.4). The function µ∗ is an outer measure.
Lemma 90 (= Lemma 12.5). If A ∈ A , then A is measurable with
respect to µ∗ .
Proposition 91 (=Proposition 12.6). Let µ be a measure on an algebra
A of subsets of a set X and let µ∗ be the outer measure induced by µ.
Then, for any E ⊂ X and any ε > 0, there is A ∈ Aσ with E ⊂ A and
µ∗ A ≤ µ∗ E + ε.
In addition, for any E ⊂ X, there is B ∈ Aσδ such that E ⊂ B and
µ∗ B = µ∗ E.
Proposition 92 (=Proposition 12.7). Let µ be a σ-finite measure on
an algebra A of subsets of a set X and let µ∗ be the outer measure
induced by µ. Then E ⊂ X is measurable if and only if there are sets
A ∈ Aσδ and B ⊂ A with µ∗ B = 0 such that E = A \ B. In addition,
if B is any subset of X with µ∗ B = 0, then there is C ∈ Aσδ with
µ∗ C = 0 and B ⊂ C.
Theorem 93 (=Theorem 12.9, Carathéodory). Let µ be a measure on
an algebra A of subsets of a set X and let µ∗ be the induced outer
measure. Then µ̄, the restriction of µ∗ to the σ-algebra B of all µ∗ measurable sets, is an extension of µ to the σ-algebra B and A ⊂ B.
If µ is finite, so is µ̄. If µ is σ-finite, so is µ̄, and µ̄ is the unique
measure on the σ-algebra generated by A which extends µ.
Proposition 94 (=Proposition 12.10). Let C be a semialgebra of subsets of a set X, and let µ be a set function on C such that µ∅ = 0 if
∅ ∈ C . Suppose also that
(1) If C ∈ C is a union
Pof finitely many disjoint elements C1 , . . . , Cn
of C , then µC =
µCi .
(2) If C ∈ C is a union
of
P countably many disjoint elements C1 , . . .
of C , then µC ≤
µCi .
Then there is a unique extension of µ to a measure on the algebra
generated by C .
THEOREMS, ETC., FOR MATH 516
13
Lemma 95 (=Lemma 12.14). Let {Ri } be a countable collection of
disjoint measurable P
rectangles such that R = ∪Ri is a measurable rectangle. Then λR =
λRi .
Lemma 96 (=Lemma 12.15). Let x ∈ X and E ∈ Rσδ . Then Ex is
measurable.
Lemma 97 (=Lemma 12.16). Let E ∈ Rσδ and suppose µ × νE < ∞.
Then g, defined by g(x) = ν(Ex ) is measurable and
Z
g dµ = µ × ν(E).
Lemma 98 (=Lemma 12.17). If µ × ν(E) = 0, then ν(Ex ) = 0 for
almost all x.
Proposition 99 (=Proposition 12.18). Let E be a measurable subset
of X × Y with µ × ν(E) < ∞. Then for almost all x, Ex is measurable,
the function g, defined by g(x) = ν(Ex ), is measurable, and
Z
g dµ = µ × νE.
In the next two theorems, if f is a function on X × Y , we define fx
and f y by fx (y) = f (x, y) and f y (x) = f (x, y).
Theorem 100 (=Theorem 12.19, Fubini). Let (X, A , µ) and (Y, B, ν)
be complete measure spaces and let f be integrable on X × Y . Then
(i)
(i)0
(ii)
(ii)0
(iii)
For almost all x, fx is integrable.
For almost all x, fx is integrable.
For almost all x, fx is integrable.
For
¢
R ¡Ralmost ¢all x, fxR is integrable.
R ¡R
f
dν
dµ
=
f
d(µ
×
ν)
=
f
dµ
dν.
X
Y
X×Y
Y
X
Theorem 101 (=Theorem 12.20, Tonelli). Let (X, A , µ) and (Y, B, ν)
be σ-finite complete measure spaces and let f be nonnegative and measurable on X × Y . Then
(i)
(i)0
(ii)
(ii)0
(iii)
For almost all x, fx is measurable.
For almost all x, fx is measurable.
For almost all x, fx is measurable.
For
¢
R ¡Ralmost ¢all x, fxR is measurable. R ¡R
f dν dµ = X×Y f d(µ × ν) = Y X f dµ dν.
Y
X
Proposition 102 (=Proposition 12.40). If µ∗ is a Carathéodory outer
measure with respect to Γ, then every function in Γ is µ∗ -measurable.
14
THEOREMS, ETC., FOR MATH 516
Proposition 103 (=Proposition 12.41). Let µ∗ be an outer measure
on a metric space (X, ρ) and suppose that, for any two subsets A and
B of X with ρ(A, B) > 0, we have µ∗ A + µ∗ B = µ∗ (A + B). Then
every closed set is measurable.