Real Analysis

Measure Theory

Preliminaries

• Lemma 1.1
If a rectangle is the almost disjoint union of finitely many other rectangles, say , then

• Lemma 1.2
If are rectangles, and , then

• Theorem 1.3
Every open subset of can be writen uniquely as a countable union of disjoint open intervals.

• Theorem 1.4
Every open subset of , , can be written as a countable union of almost disjoint closed cubes.

The exterior measure

• Definition
The precise definition is as follows: if is any subset of , the exterior measure of is

Measurable sets and the Lebesgue measure

• Definition
A subset of is Lebesgue measurable, or simply measurable, if for any there exists an open set with and
If E is measurable, we define its Lebesgue measure (or measure) by

• Property 1
Every open set in is measurable.

• Property 2
If , then is measurable. In particular, if is a subset of a set of exterior measure , then is measurable.

• Property 3
A countable union of measurable sets is measurable.

• Property 4
Closed sets are measurable.

• Lemma 3.1
If is closed, is compact, and these sets are disjoint, then .

• Property 5
The complement of a measurable set is measurable.

• Property 6
A countable intersection of measurable sets is measurable.

• Theorem 3.2
If , are disjoint measurable sets, and , then

• Definition
If is a countable collection of subsets of that increases to in the sense that for all , and , then we write . Similarly, if decreases to in the sense that for all , and , we write .

• Corollary 3.3
Suppose are measurable subsets of .
(i) If , then .
(ii) If and for some , then

• Theorem 3.4
Suppose is a measurable subset of . Then, for every :
(i) There exists an open set with and .
(ii) There exists a closed set with and .
(iii) If is finite, there exists a compact set with and .
(iv) If is finite, there exists a finite union of closed cubes such that .

• Invariance properties of Lebesgue measure
If is a measurable set and , then the set is also measurable, and .
Suppose , and denote by the set . We can then assert that is measurable whenever is, and .
Whenever is measurable, so is and .

• Definition
A σ-algebra of sets is a collection of subsets of that is closed under countable unions, countable intersections, and complements.
Another σ-algebra, which plays a vital role in analysis, is the Borel σ-algebra in , denoted by , which by definition is the smallest σ-algebra that contains all open sets. Elements of this σ-algebra are called Borel sets.
Starting with the open and closed sets, which are the simplest Borel sets, one could try to list the Borel sets in order of their complexity. Next in order would come countable intersections of open sets; such sets are called sets. Alternatively, one could consider their complements, the countable union of closed sets, called the sets.

• Corollary 3.5
A subset of is measurable
(i) if and only if differs from a by a set of measure zero,
(ii) if and only if differs from an by a set of measure zero.

• Construction of a non-measurable set
 

Measurable functions

• Definition
A function defined on a measurable subset of is measurable, if for all , the set is measurable.

• Property 1
The finite-valued function is measurable if and only if is measurable for every open set , and if and only if is measurable for every closed set .

• Property 2
If is continuous on , then is measurable. If is measurable and finite-valued, and is continuous, then is measurable.

• Property 3
Suppose is a sequence of measurable functions. Then
are measurable.

• Property 4
If is a collection of measurable functions, and
then is measurable.

• Property 5
If and are measurable, then
1. The integer powers are measurable.
2. and are measurable if both and are finite-valued.

• Property 6
Suppose is measurable, and for . Then is measurable.

• Theorem 4.1
Suppose is a non-negative measurable function on . Then there exists an increasing sequence of non-negative simple functions that converges pointwise to , namely,

• Theorem 4.2
Suppose is measurable on . Then there exists a sequence of simple functions that satisfies
In particular, we have for all and .

• Theorem 4.3
Suppose is measurable on . Then there exists a sequence of step functions that converges pointwise to for almost every .

• Theorem 4.4 (Egorov)
Suppose is a sequence of measurable functions defined on a measurable set with , and assume that on . Given , we can find a closed set such that and uniformly on .

• Theorem 4.5 (Lusin)
Suppose is measurable and finite valued on with of finite measure. Then for every there exists a closed set , with
and such that is continuous.

Integration Theory

The Lebesgue integral: basic properties and conver- gence theorems

Stage one: simple functions

• Definition
A simple function is a finite sum
where the are measurable sets of finite measure and the are constants.
The canonical form of is the unique decomposition as in (1), where the numbers are distinct and non-zero, and the sets are disjoint.

• Definition
If is a simple function with canonical form , then we define the Lebesgue integral of by
If is a measurable subset of with finite measure, then is also a simple function, and we define
To emphasize the choice of the Lebesgue measure m in the definition of the integral, one sometimes writes
for the Lebesgue integral of . In fact, as a matter of convenience, we shall often write or simply for the integral of over .

• Proposition 1.1
The integral of simple functions defined above satisfies the following properties:
1. Independence of the representation. If is any representation of , then
2. Linearity. If and are simple, and , then
3. Additivity. If and are disjoint subsets of with finite measure, then
4. Monotonicity. If are simple, then
5. Triangle inequality. If is a simple function, then so is , and

Stage two: bounded functions supported on a set of finite measure

• Definition
The support of a measurable function is defined to be the set of all points where does not vanish,
We shall also say that is supported on a set , if whenever .

• Lemma 1.2
Let be a bounded function supported on a set of finite measure. If is any sequence of simple functions bounded by , supported on , and with for , then:
1. The limit exists.
2. If , then the limit equals .

• Definition
Using Lemma 1.2 we can now turn to the integration of bounded functions that are supported on sets of finite measure. For such a function we define its Lebesgue integral by
where is any sequence of simple functions satisfying: , each is supported on the support of , and for as tends to infinity. By the previous lemma, we know that this limit exists.

• 切比雪夫不等式
设 肺腑可测函数，任给 ，则

• Proposition 1.3
Suppose and are bounded functions supported on sets of finite measure. Then the following properties hold.
1. Linearity. If , then
2. Additivity. If and are disjoint subsets of , then
3. Monotonicity. If are simple, then
4. Triangle inequality. is also bounded, supported on a set of finite measure, and

• Theorem 1.4 (Bounded convergence theorem)
Suppose that is a sequence of measurable functions that are all bounded by , are supported on a set of finite measure, and as . Then is measurable, bounded, supported on for , and
Consequently,

• Theorem 1.5
Suppose is Riemann integrable on the closed interval . Then is measurable, and