Lebesgue Measure and Lebesgue Integral - Real Analysis, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Page 1


Chapter 1
Lebesgue Measure and Integration
We will give a brief review, without proofs, of Lebesgue measure and inte-
gration on subsets of R
d
. Details and proofs can be found in texts on real
analysis, such as [Fol99], [WZ77], [SS05], or [Heil18]. While some proofs are
quite easy, other proofs are surprisingly di?cult, and there are many coun-
terintuitive issues related to Lebesgue measure and integration.
1.1 Exterior Lebesgue Measure
For compactness of notation, we will refer to rectangular parallelepipeds in
R
d
whose sides are parallel to the coordinate axes simply as “boxes.”
De?nition 1.1.1. (a) A box inR
d
is a set of the form
Q = [a
1
,b
1
]×···×[a
d
,b
d
] =
d
Y
i=1
[a
i
,b
i
].
The volume of this box is
vol(Q) = (b
1
-a
1
)···(b
d
-a
d
) =
d
Y
i=1
(b
i
-a
i
).
(b) Theexterior Lebesgue measure orouter Lebesgue measure ofasetE?R
d
is
|E|
e
= inf
n
X
k
vol(Q
k
)
o
,
wherethein?mumistakenoverall ?nite or countable collectionsofboxes
Q
k
such that E ?
S
k
Q
k
. ?
Page 2


Chapter 1
Lebesgue Measure and Integration
We will give a brief review, without proofs, of Lebesgue measure and inte-
gration on subsets of R
d
. Details and proofs can be found in texts on real
analysis, such as [Fol99], [WZ77], [SS05], or [Heil18]. While some proofs are
quite easy, other proofs are surprisingly di?cult, and there are many coun-
terintuitive issues related to Lebesgue measure and integration.
1.1 Exterior Lebesgue Measure
For compactness of notation, we will refer to rectangular parallelepipeds in
R
d
whose sides are parallel to the coordinate axes simply as “boxes.”
De?nition 1.1.1. (a) A box inR
d
is a set of the form
Q = [a
1
,b
1
]×···×[a
d
,b
d
] =
d
Y
i=1
[a
i
,b
i
].
The volume of this box is
vol(Q) = (b
1
-a
1
)···(b
d
-a
d
) =
d
Y
i=1
(b
i
-a
i
).
(b) Theexterior Lebesgue measure orouter Lebesgue measure ofasetE?R
d
is
|E|
e
= inf
n
X
k
vol(Q
k
)
o
,
wherethein?mumistakenoverall ?nite or countable collectionsofboxes
Q
k
such that E ?
S
k
Q
k
. ?
Thus, every subset ofR
d
has a uniquely de?ned exterior measure that lies
in the range 0=|E|
e
=8. Here are some of the basic properties of exterior
measure.
Theorem 1.1.2. (a) If Q is a box in R
d
, then |Q|
e
= vol(Q).
(b) Monotonicity: If E ?F ?R
d
, then |E|
e
=|F|
e
.
(c) Countable subadditivity: If E
k
?R
d
for k?N, then
¯
¯
¯
8
S
k=1
E
k
¯
¯
¯
e
=
8
X
k=1
|E
k
|
e
.
(d) Translation invariance: If E ? R
d
and h ? R
d
, then |E +h|
e
= |E|
e
,
where E +h ={t+h :t?E}.
(e) Regularity: If E ? R
d
and e > 0, then there exists an open set U ? E
such that |U|
e
=|E|
e
+e, and hence
|E|
e
= inf
©
|U|
e
:U open, U ?E
ª
. ?
1.2 Lebesgue Measure
De?nition 1.2.1. A set E ?R
d
is Lebesgue measurable, or simply measur-
able, if
?e> 0, ? open U ?E such that|U\E|
e
=e. (1.1)
If E is Lebesgue measurable, then its Lebesgue measure is its exterior
Lebesgue measure and is denoted by|E| =|E|
e
. ?
Note that equation (1.1) does not follow from Theorem 1.1.2(e). One con-
sequence of the Axiom of Choice is that there exist subsets of R
d
that are
not measurable.
Thefollowingresultsummarizessomeofthepropertiesofmeasurablesets.
Theorem 1.2.2. (a) The class of measurable subsets of R
d
is a s-algebra,
meaning that:
i.Ø and R
d
are measurable,
ii. if E
1
,E
2
,... are measurable, then ?E
k
is measurable,
iii. if E is measurable, then R
d
\E is measurable.
(b) Every open and every closed subset of R
d
is measurable.
(c) Every subset E of R
d
with |E|
e
= 0 is measurable. ?
Sincemeasurabilityispreservedundercomplementsandcountableunions,
it is also preserved under countable intersections.
We give some equivalent formulations of measurability.
Page 3


Chapter 1
Lebesgue Measure and Integration
We will give a brief review, without proofs, of Lebesgue measure and inte-
gration on subsets of R
d
. Details and proofs can be found in texts on real
analysis, such as [Fol99], [WZ77], [SS05], or [Heil18]. While some proofs are
quite easy, other proofs are surprisingly di?cult, and there are many coun-
terintuitive issues related to Lebesgue measure and integration.
1.1 Exterior Lebesgue Measure
For compactness of notation, we will refer to rectangular parallelepipeds in
R
d
whose sides are parallel to the coordinate axes simply as “boxes.”
De?nition 1.1.1. (a) A box inR
d
is a set of the form
Q = [a
1
,b
1
]×···×[a
d
,b
d
] =
d
Y
i=1
[a
i
,b
i
].
The volume of this box is
vol(Q) = (b
1
-a
1
)···(b
d
-a
d
) =
d
Y
i=1
(b
i
-a
i
).
(b) Theexterior Lebesgue measure orouter Lebesgue measure ofasetE?R
d
is
|E|
e
= inf
n
X
k
vol(Q
k
)
o
,
wherethein?mumistakenoverall ?nite or countable collectionsofboxes
Q
k
such that E ?
S
k
Q
k
. ?
Thus, every subset ofR
d
has a uniquely de?ned exterior measure that lies
in the range 0=|E|
e
=8. Here are some of the basic properties of exterior
measure.
Theorem 1.1.2. (a) If Q is a box in R
d
, then |Q|
e
= vol(Q).
(b) Monotonicity: If E ?F ?R
d
, then |E|
e
=|F|
e
.
(c) Countable subadditivity: If E
k
?R
d
for k?N, then
¯
¯
¯
8
S
k=1
E
k
¯
¯
¯
e
=
8
X
k=1
|E
k
|
e
.
(d) Translation invariance: If E ? R
d
and h ? R
d
, then |E +h|
e
= |E|
e
,
where E +h ={t+h :t?E}.
(e) Regularity: If E ? R
d
and e > 0, then there exists an open set U ? E
such that |U|
e
=|E|
e
+e, and hence
|E|
e
= inf
©
|U|
e
:U open, U ?E
ª
. ?
1.2 Lebesgue Measure
De?nition 1.2.1. A set E ?R
d
is Lebesgue measurable, or simply measur-
able, if
?e> 0, ? open U ?E such that|U\E|
e
=e. (1.1)
If E is Lebesgue measurable, then its Lebesgue measure is its exterior
Lebesgue measure and is denoted by|E| =|E|
e
. ?
Note that equation (1.1) does not follow from Theorem 1.1.2(e). One con-
sequence of the Axiom of Choice is that there exist subsets of R
d
that are
not measurable.
Thefollowingresultsummarizessomeofthepropertiesofmeasurablesets.
Theorem 1.2.2. (a) The class of measurable subsets of R
d
is a s-algebra,
meaning that:
i.Ø and R
d
are measurable,
ii. if E
1
,E
2
,... are measurable, then ?E
k
is measurable,
iii. if E is measurable, then R
d
\E is measurable.
(b) Every open and every closed subset of R
d
is measurable.
(c) Every subset E of R
d
with |E|
e
= 0 is measurable. ?
Sincemeasurabilityispreservedundercomplementsandcountableunions,
it is also preserved under countable intersections.
We give some equivalent formulations of measurability.
1.2 Lebesgue Measure
De?nition 1.2.3. (a) A set H ? R
d
is a G
d
-set if there exist ?nitely or
countably many open sets U
k
such that H =nU
k
.
(b) A setH ?R
d
is anF
s
-set if there exist ?nitely or countably many closed
sets F
k
such that H =?F
k
. ?
Theorem 1.2.4. Let E ? R
d
be given. Then the following statements are
equivalent.
(a)E is measurable.
(b) For every e> 0, there exists a closed set F ?E such that |E\F|
e
=e.
(c)E =H\Z where H is a G
d
-set and |Z| = 0.
(d)E =H?Z where H is an F
s
-set and |Z| = 0. ?
Next we list some properties of Lebesgue measure.
Theorem 1.2.5. Let E and E
k
for k?N be measurable subsets of R
d
.
(a) Countable additivity: If E
1
, E
2
,... are disjoint measurable subsets ofR
d
,
then
¯
¯
¯
8
S
k=1
E
k
¯
¯
¯ =
8
X
k=1
|E
k
|.
(b) If E
1
?E
2
and |E
1
|<8, then |E
2
\E
1
| =|E
2
|-|E
1
|.
(c) Continuity from below: If E
1
?E
2
?··· , then
¯
¯
?E
k
¯
¯
= lim
k?8
|E
k
|.
(d) Continuity from above: If E
1
? E
2
? ··· and |E
1
| < 8, then
¯
¯
nE
k
¯
¯
=
lim
k?8
|E
k
|.
(e) Translation invariance: If h ? R
d
, then |E +h| = |E|, where E +h =
{x+h :x?E}.
(f) Linear changes of variable: If T: R
d
? R
d
is linear, then T(E) is mea-
surable and |T(E)| =|det(T)||E|.
(g) Cartesianproducts: IfE?R
m
andF ?R
n
are measurable, thenE×F ?
R
m+n
is measurable and |E ×F| = |E||F| (using the convention that
0·8 = 0). ?
We end this section with some terminology.
De?nition 1.2.6. A property that holds except possibly on a set of measure
zero is said to hold almost everywhere, abbreviated a.e. ?
For example, if C is the classical Cantor middle-thirds set, then |C| = 0.
Hence, the characteristic function
?
C
of C satis?es
?
C
(t) = 0 except for
thosetthatbelongtothezeromeasuresetC.Thereforewesaythat
?
C
(t) = 0
for almost every t, or
?
C
= 0 a.e. for short.
The essential supremum of a function is an example of a quantity that is
de?ned in terms of a property that holds almost everywhere.
Page 4


Chapter 1
Lebesgue Measure and Integration
We will give a brief review, without proofs, of Lebesgue measure and inte-
gration on subsets of R
d
. Details and proofs can be found in texts on real
analysis, such as [Fol99], [WZ77], [SS05], or [Heil18]. While some proofs are
quite easy, other proofs are surprisingly di?cult, and there are many coun-
terintuitive issues related to Lebesgue measure and integration.
1.1 Exterior Lebesgue Measure
For compactness of notation, we will refer to rectangular parallelepipeds in
R
d
whose sides are parallel to the coordinate axes simply as “boxes.”
De?nition 1.1.1. (a) A box inR
d
is a set of the form
Q = [a
1
,b
1
]×···×[a
d
,b
d
] =
d
Y
i=1
[a
i
,b
i
].
The volume of this box is
vol(Q) = (b
1
-a
1
)···(b
d
-a
d
) =
d
Y
i=1
(b
i
-a
i
).
(b) Theexterior Lebesgue measure orouter Lebesgue measure ofasetE?R
d
is
|E|
e
= inf
n
X
k
vol(Q
k
)
o
,
wherethein?mumistakenoverall ?nite or countable collectionsofboxes
Q
k
such that E ?
S
k
Q
k
. ?
Thus, every subset ofR
d
has a uniquely de?ned exterior measure that lies
in the range 0=|E|
e
=8. Here are some of the basic properties of exterior
measure.
Theorem 1.1.2. (a) If Q is a box in R
d
, then |Q|
e
= vol(Q).
(b) Monotonicity: If E ?F ?R
d
, then |E|
e
=|F|
e
.
(c) Countable subadditivity: If E
k
?R
d
for k?N, then
¯
¯
¯
8
S
k=1
E
k
¯
¯
¯
e
=
8
X
k=1
|E
k
|
e
.
(d) Translation invariance: If E ? R
d
and h ? R
d
, then |E +h|
e
= |E|
e
,
where E +h ={t+h :t?E}.
(e) Regularity: If E ? R
d
and e > 0, then there exists an open set U ? E
such that |U|
e
=|E|
e
+e, and hence
|E|
e
= inf
©
|U|
e
:U open, U ?E
ª
. ?
1.2 Lebesgue Measure
De?nition 1.2.1. A set E ?R
d
is Lebesgue measurable, or simply measur-
able, if
?e> 0, ? open U ?E such that|U\E|
e
=e. (1.1)
If E is Lebesgue measurable, then its Lebesgue measure is its exterior
Lebesgue measure and is denoted by|E| =|E|
e
. ?
Note that equation (1.1) does not follow from Theorem 1.1.2(e). One con-
sequence of the Axiom of Choice is that there exist subsets of R
d
that are
not measurable.
Thefollowingresultsummarizessomeofthepropertiesofmeasurablesets.
Theorem 1.2.2. (a) The class of measurable subsets of R
d
is a s-algebra,
meaning that:
i.Ø and R
d
are measurable,
ii. if E
1
,E
2
,... are measurable, then ?E
k
is measurable,
iii. if E is measurable, then R
d
\E is measurable.
(b) Every open and every closed subset of R
d
is measurable.
(c) Every subset E of R
d
with |E|
e
= 0 is measurable. ?
Sincemeasurabilityispreservedundercomplementsandcountableunions,
it is also preserved under countable intersections.
We give some equivalent formulations of measurability.
1.2 Lebesgue Measure
De?nition 1.2.3. (a) A set H ? R
d
is a G
d
-set if there exist ?nitely or
countably many open sets U
k
such that H =nU
k
.
(b) A setH ?R
d
is anF
s
-set if there exist ?nitely or countably many closed
sets F
k
such that H =?F
k
. ?
Theorem 1.2.4. Let E ? R
d
be given. Then the following statements are
equivalent.
(a)E is measurable.
(b) For every e> 0, there exists a closed set F ?E such that |E\F|
e
=e.
(c)E =H\Z where H is a G
d
-set and |Z| = 0.
(d)E =H?Z where H is an F
s
-set and |Z| = 0. ?
Next we list some properties of Lebesgue measure.
Theorem 1.2.5. Let E and E
k
for k?N be measurable subsets of R
d
.
(a) Countable additivity: If E
1
, E
2
,... are disjoint measurable subsets ofR
d
,
then
¯
¯
¯
8
S
k=1
E
k
¯
¯
¯ =
8
X
k=1
|E
k
|.
(b) If E
1
?E
2
and |E
1
|<8, then |E
2
\E
1
| =|E
2
|-|E
1
|.
(c) Continuity from below: If E
1
?E
2
?··· , then
¯
¯
?E
k
¯
¯
= lim
k?8
|E
k
|.
(d) Continuity from above: If E
1
? E
2
? ··· and |E
1
| < 8, then
¯
¯
nE
k
¯
¯
=
lim
k?8
|E
k
|.
(e) Translation invariance: If h ? R
d
, then |E +h| = |E|, where E +h =
{x+h :x?E}.
(f) Linear changes of variable: If T: R
d
? R
d
is linear, then T(E) is mea-
surable and |T(E)| =|det(T)||E|.
(g) Cartesianproducts: IfE?R
m
andF ?R
n
are measurable, thenE×F ?
R
m+n
is measurable and |E ×F| = |E||F| (using the convention that
0·8 = 0). ?
We end this section with some terminology.
De?nition 1.2.6. A property that holds except possibly on a set of measure
zero is said to hold almost everywhere, abbreviated a.e. ?
For example, if C is the classical Cantor middle-thirds set, then |C| = 0.
Hence, the characteristic function
?
C
of C satis?es
?
C
(t) = 0 except for
thosetthatbelongtothezeromeasuresetC.Thereforewesaythat
?
C
(t) = 0
for almost every t, or
?
C
= 0 a.e. for short.
The essential supremum of a function is an example of a quantity that is
de?ned in terms of a property that holds almost everywhere.
De?nition 1.2.7(Essential Supremum). The essential supremum of a
function f: E?R is
esssup
t?E
f(t) = inf{M :f =M a.e.}.
We say that f is essentially bounded if esssup
t?E
|f(t)|<8. ?
1.3 Measurable Functions
Now we de?ne the class of measurable functions on subsets ofR
d
.
De?nition 1.3.1(Real-Valued Measurable Functions). Fix a measur-
ablesetE ?R
d
,andletf: E ?Rbegiven.Thenf isa Lebesgue measurable
function, or simply a measurable function, if f
-1
(a,8) ={t?E :f(t)>a}
is a measurable subset ofR
d
for each a?R. ?
Inparticular,everycontinuousfunctionf:R
d
?Rismeasurable,because
theinverseimageofanopensetunderacontinuousfunctionisopen.However,
a measurable function need not be continuous.
Measurability is preserved under most of the usual operations, including
addition, multiplication, and limits. Some care does need to be taken with
compositions, but if we compose a measurable function with a continuous
function in the correct order, then measurability will be assured.
Theorem 1.3.2. Let E?R
d
be measurable.
(a) If f: E?R is measurable and g =f a.e., then g is measurable.
(b) If f, g: E?R are measurable, then so is f +g.
(c) If f: E ? R is measurable and ?: R ? R is continuous, then ??f is
measurable. Consequently, |f|, f
2
, f
+
, f
-
, and |f|
p
for p > 0 are all
measurable.
(d) If f, g: E?R are measurable, then so is fg.
(e) If f
n
: E ? R are measurable for n ? N, then so are supf
n
, inff
n
,
limsupf
n
, and liminff
n
.
(f) If f
n
: E ?R are measurable for n?N and f(t) = lim
n?8
f
n
(t) exists
for a.e. t, then f is measurable. ?
De?nition 1.3.3(Complex-Valued Measurable Functions). Given a
measurable domain E ? R
d
and a complex-valued function f: E ? C,
write f in real and imaginary parts as f = f
r
+if
i
. Then we say that f
is measurable if both f
r
and f
i
are measurable. ?
Egoro?’s Theorem says that pointwise convergence of measurable func-
tions is uniform convergence on “most” of the set.
Page 5


Chapter 1
Lebesgue Measure and Integration
We will give a brief review, without proofs, of Lebesgue measure and inte-
gration on subsets of R
d
. Details and proofs can be found in texts on real
analysis, such as [Fol99], [WZ77], [SS05], or [Heil18]. While some proofs are
quite easy, other proofs are surprisingly di?cult, and there are many coun-
terintuitive issues related to Lebesgue measure and integration.
1.1 Exterior Lebesgue Measure
For compactness of notation, we will refer to rectangular parallelepipeds in
R
d
whose sides are parallel to the coordinate axes simply as “boxes.”
De?nition 1.1.1. (a) A box inR
d
is a set of the form
Q = [a
1
,b
1
]×···×[a
d
,b
d
] =
d
Y
i=1
[a
i
,b
i
].
The volume of this box is
vol(Q) = (b
1
-a
1
)···(b
d
-a
d
) =
d
Y
i=1
(b
i
-a
i
).
(b) Theexterior Lebesgue measure orouter Lebesgue measure ofasetE?R
d
is
|E|
e
= inf
n
X
k
vol(Q
k
)
o
,
wherethein?mumistakenoverall ?nite or countable collectionsofboxes
Q
k
such that E ?
S
k
Q
k
. ?
Thus, every subset ofR
d
has a uniquely de?ned exterior measure that lies
in the range 0=|E|
e
=8. Here are some of the basic properties of exterior
measure.
Theorem 1.1.2. (a) If Q is a box in R
d
, then |Q|
e
= vol(Q).
(b) Monotonicity: If E ?F ?R
d
, then |E|
e
=|F|
e
.
(c) Countable subadditivity: If E
k
?R
d
for k?N, then
¯
¯
¯
8
S
k=1
E
k
¯
¯
¯
e
=
8
X
k=1
|E
k
|
e
.
(d) Translation invariance: If E ? R
d
and h ? R
d
, then |E +h|
e
= |E|
e
,
where E +h ={t+h :t?E}.
(e) Regularity: If E ? R
d
and e > 0, then there exists an open set U ? E
such that |U|
e
=|E|
e
+e, and hence
|E|
e
= inf
©
|U|
e
:U open, U ?E
ª
. ?
1.2 Lebesgue Measure
De?nition 1.2.1. A set E ?R
d
is Lebesgue measurable, or simply measur-
able, if
?e> 0, ? open U ?E such that|U\E|
e
=e. (1.1)
If E is Lebesgue measurable, then its Lebesgue measure is its exterior
Lebesgue measure and is denoted by|E| =|E|
e
. ?
Note that equation (1.1) does not follow from Theorem 1.1.2(e). One con-
sequence of the Axiom of Choice is that there exist subsets of R
d
that are
not measurable.
Thefollowingresultsummarizessomeofthepropertiesofmeasurablesets.
Theorem 1.2.2. (a) The class of measurable subsets of R
d
is a s-algebra,
meaning that:
i.Ø and R
d
are measurable,
ii. if E
1
,E
2
,... are measurable, then ?E
k
is measurable,
iii. if E is measurable, then R
d
\E is measurable.
(b) Every open and every closed subset of R
d
is measurable.
(c) Every subset E of R
d
with |E|
e
= 0 is measurable. ?
Sincemeasurabilityispreservedundercomplementsandcountableunions,
it is also preserved under countable intersections.
We give some equivalent formulations of measurability.
1.2 Lebesgue Measure
De?nition 1.2.3. (a) A set H ? R
d
is a G
d
-set if there exist ?nitely or
countably many open sets U
k
such that H =nU
k
.
(b) A setH ?R
d
is anF
s
-set if there exist ?nitely or countably many closed
sets F
k
such that H =?F
k
. ?
Theorem 1.2.4. Let E ? R
d
be given. Then the following statements are
equivalent.
(a)E is measurable.
(b) For every e> 0, there exists a closed set F ?E such that |E\F|
e
=e.
(c)E =H\Z where H is a G
d
-set and |Z| = 0.
(d)E =H?Z where H is an F
s
-set and |Z| = 0. ?
Next we list some properties of Lebesgue measure.
Theorem 1.2.5. Let E and E
k
for k?N be measurable subsets of R
d
.
(a) Countable additivity: If E
1
, E
2
,... are disjoint measurable subsets ofR
d
,
then
¯
¯
¯
8
S
k=1
E
k
¯
¯
¯ =
8
X
k=1
|E
k
|.
(b) If E
1
?E
2
and |E
1
|<8, then |E
2
\E
1
| =|E
2
|-|E
1
|.
(c) Continuity from below: If E
1
?E
2
?··· , then
¯
¯
?E
k
¯
¯
= lim
k?8
|E
k
|.
(d) Continuity from above: If E
1
? E
2
? ··· and |E
1
| < 8, then
¯
¯
nE
k
¯
¯
=
lim
k?8
|E
k
|.
(e) Translation invariance: If h ? R
d
, then |E +h| = |E|, where E +h =
{x+h :x?E}.
(f) Linear changes of variable: If T: R
d
? R
d
is linear, then T(E) is mea-
surable and |T(E)| =|det(T)||E|.
(g) Cartesianproducts: IfE?R
m
andF ?R
n
are measurable, thenE×F ?
R
m+n
is measurable and |E ×F| = |E||F| (using the convention that
0·8 = 0). ?
We end this section with some terminology.
De?nition 1.2.6. A property that holds except possibly on a set of measure
zero is said to hold almost everywhere, abbreviated a.e. ?
For example, if C is the classical Cantor middle-thirds set, then |C| = 0.
Hence, the characteristic function
?
C
of C satis?es
?
C
(t) = 0 except for
thosetthatbelongtothezeromeasuresetC.Thereforewesaythat
?
C
(t) = 0
for almost every t, or
?
C
= 0 a.e. for short.
The essential supremum of a function is an example of a quantity that is
de?ned in terms of a property that holds almost everywhere.
De?nition 1.2.7(Essential Supremum). The essential supremum of a
function f: E?R is
esssup
t?E
f(t) = inf{M :f =M a.e.}.
We say that f is essentially bounded if esssup
t?E
|f(t)|<8. ?
1.3 Measurable Functions
Now we de?ne the class of measurable functions on subsets ofR
d
.
De?nition 1.3.1(Real-Valued Measurable Functions). Fix a measur-
ablesetE ?R
d
,andletf: E ?Rbegiven.Thenf isa Lebesgue measurable
function, or simply a measurable function, if f
-1
(a,8) ={t?E :f(t)>a}
is a measurable subset ofR
d
for each a?R. ?
Inparticular,everycontinuousfunctionf:R
d
?Rismeasurable,because
theinverseimageofanopensetunderacontinuousfunctionisopen.However,
a measurable function need not be continuous.
Measurability is preserved under most of the usual operations, including
addition, multiplication, and limits. Some care does need to be taken with
compositions, but if we compose a measurable function with a continuous
function in the correct order, then measurability will be assured.
Theorem 1.3.2. Let E?R
d
be measurable.
(a) If f: E?R is measurable and g =f a.e., then g is measurable.
(b) If f, g: E?R are measurable, then so is f +g.
(c) If f: E ? R is measurable and ?: R ? R is continuous, then ??f is
measurable. Consequently, |f|, f
2
, f
+
, f
-
, and |f|
p
for p > 0 are all
measurable.
(d) If f, g: E?R are measurable, then so is fg.
(e) If f
n
: E ? R are measurable for n ? N, then so are supf
n
, inff
n
,
limsupf
n
, and liminff
n
.
(f) If f
n
: E ?R are measurable for n?N and f(t) = lim
n?8
f
n
(t) exists
for a.e. t, then f is measurable. ?
De?nition 1.3.3(Complex-Valued Measurable Functions). Given a
measurable domain E ? R
d
and a complex-valued function f: E ? C,
write f in real and imaginary parts as f = f
r
+if
i
. Then we say that f
is measurable if both f
r
and f
i
are measurable. ?
Egoro?’s Theorem says that pointwise convergence of measurable func-
tions is uniform convergence on “most” of the set.
Theorem 1.3.4(Egoro?’s Theorem). Let E ? R
d
be measurable with
|E| < 8. If f
n
, f: E ? C are measurable functions and f
n
(t) ? f(t) for
a.e. t ? E, then for every e > 0 there exists a measurable set A ? E such
that |A|<e and f
n
converges uniformly to f on E\A, i.e.,
lim
n?8
µ
sup
t/ ?A
|f(t)-f
n
(t)|
¶
= 0. ?
1.4 The Lebesgue Integral
To de?ne the Lebesgue integral of a measurable function, we ?rst begin with
“simple functions” and then extend to nonnegative functions, real-valued
functions, and complex-valued functions.
De?nition 1.4.1. Let E ?R
d
be measurable.
(a) A simple function on E is a measurable function f: E ? F that takes
only ?nitely many distinct values. That is, f: E?F is simple if
f =
N
X
k=1
a
k
?
E
k
, (1.2)
where N > 0, a
k
?F, and the E
k
are measurable subsets of E.
(b) If a
1
,...,a
N
?F are the distinct values assumed by a simple function f
and we set E
k
= {t ? E : f(t) = a
k
}, then f has the form given in
equation (1.2) and the setsE
1
,...,E
N
form a partition ofE. We call this
the standard representation of f.
(c) If f is a nonnegative simple function on E with standard representation
f =
P
N
k=1
a
k
?
E
k
, then the Lebesgue integral of f over E is
Z
E
f =
Z
E
f(t)dt =
N
X
k=1
a
k
|E
k
|.
(d) Iff: E? [0,8) is a measurable function, then the Lebesgue integral off
over E is
Z
E
f =
Z
E
f(t)dt = sup
½Z
E
f : 0=f=f, f simple
¾
.
If A is a measurable subset of E, then we write
R
A
f =
R
E
f
?
A
. ?
Followingaresomeofthebasicpropertiesofintegralsofnonnegativefunc-
tions.