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Lebesgue Integration

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Integration of Simple Functions

Lebesgue Integral of a Simple Function

If is a simple function with canonical form then the Lebesgue integral of is defined as

If is a measurable subset of with finite measure, then is also a simple function, and we define

e.g. Integral of a characteristic function is the measure of the set.

Properties of the Integral of Simple Functions

The integral of simple functions satisfies the following properties:

  1. Independence of the representation: If is any representation of , then
  2. Linearity: If and are simple functions, and , then
  3. Additivity: If and are disjoint subsets of with finite measure, then
  4. Monotonicity: If are simple functions, then
  5. Triangle inequality: If is a simple function, then so is , and

Proposition

If and are a pair of simple functions that agree almost everywhere, then

This identity of integrals for functions that agree almost everywhere will continue to hold for successive definitions of the integral.

Bounded Functions Supported on a Set of Finite Measure

Support

Suppose that is a real valued function whose domain is an arbitrary set . The support of , written , is the set of points in where is non-zero:If for all but a finite number of points in , then is said to have finite support. We shall also say that is supported on a set , if whenever .

is measurable then is a measurable set.

If

Proposition

If is a measurable function bounded by and supported on a set , then there exists a sequence of simple functions, with each bounded by and supported on , and such that

Proof Since is bounded by , is a non-negative function. The theorem states that there exists an increasing sequence of non-negative simple functions converging pointwise to . Define . Then is a sequence of simple functions bounded by and converging pointwise to .

The following lemma allows us to define the integral for bounded functions supported on sets of finite measure:

Lemma

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 a.e. , then:

  1. The limit exists.
  2. If a.e., then the limit equals .

Lebesgue Integral of a Bounded Function Supported on Sets of Finite Measure

Let be a measurable function bounded by , supported on a set of finite measure. We define the Lebesgue integral of as where is a sequence of simple functions bounded by , , and converging pointwise to a.e.

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 for a.e. as . Then is measurable, bounded, supported on for a.e. , and

Consequently,

Proof Fix any , by Egorov’s theorem, we can find a closed set such that and uniformly on , so there exists such that for all and . Then we have for all . Since is arbitrary, we have .

Remark

The Bounded Convergence Theorem is about interchange of limits and integrals. It says, under certain conditions, we can interchange the limit and the integral:

Proposition

If non-negative is bounded and supported on a set of finite measure and , then a.e.

Proof For each integer , we have this implies by monotonicity. Thus for all , and since , we have .

Relationship with Riemann Integral

Theorem

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

where the integral on the left-hand side is the standard Riemann integral, and that on the right-hand side is the Lebesgue integral.

Non-negative Measurable Functions

Extended Lebesgue Integral of a Non-negative Measurable Function

Suppose is a non-negative measurable function, which is not necessarily bounded. We define the extended Lebesgue integral of as where the supremum is taken over all non-negative measurable functions that are bounded and supported on a set of finite measure. If the supremum is finite, then we say that is Lebesgue integrable. Clearly if is any measurable set, then we define

Properties of the Integral of Non-Negative Measurable Functions

The integral of non-negative measurable functions enjoys the following properties:

  1. Linearity: If , and are positive real numbers, then
  2. Additivity: If and are disjoint subsets of , and , then
  3. Monotonicity: If , then
  4. Triangle inequality: If is a non-negative measurable function, then so is , and
  5. If is integrable and , then is integrable.
  6. If is integrable, then for almost all .
  7. If , then for a.e. .

Note that and for a.e. not necessarily imply . For example, consider Then as for every , but for all . Indeed, the limit of the integrals is greater than the integral of the limit function in general:

Fatou's Lemma

Suppose is a sequence of measurable functions with . If for a.e. , then

Proof Suppose , where is bounded and supported on a set of finite measure. And we let , then is measurable, supported on , and for a.e. . By the bounded convergence theorem, we have By construction, , so that for all , and thereforeby the definition of the limit inferior. Since is arbitrary, taking the supremum over all yields .

Monotone Convergence Theorem

Suppose is a non-negative measurable function, and is a sequence of non-negative measurable functions, and for almost every . Then

Proof Since for a.e. , by monotonicity, we have , so . Hence, it follows that .

Comment

To simplify our notation, we shall write to mean that converges to in a nondecreasing sense, and to mean that converges to in a nonincreasing sense.

Corollary

Consider a series , where is a measurable function for every . Then

If is finite, then the series converges for a.e. .

Proof Let and . Then and are measurable, and , by the monotone convergence theorem, we have If is finite, then is integrable, so is finite for a.e. by the property 5.

General Case

Lebesgue Integral of a Real-Valued Measurable Function

A real-valued measurable function on is said to be Lebesgue integrable if the non-negative measurable function is integrable, i.e. . If is Lebesgue integrable, its integral is defined as follows. We first define the positive and negative parts of by

so that both and are non-negative and satisfy

Both functions and are integrable whenever is. We then define the Lebesgue integral of by

Proposition

The integral of Lebesgue integrable functions is independent of the decomposition.

Proposition

The integral of Lebesgue integrable functions is linear, additive, monotonic, and satisfies the triangle inequality.

Lemma

Suppose is integrable on . Then for every :

  1. Existence of a finite measure set: There exists a set of finite measure (a ball, for example) such that

  2. Absolute continuity of the integral: There exists a such that

Dominated Convergence Theorem

Suppose is a sequence of measurable functions such that a.e. as . If , where is integrable, then

and consequently,

Proof For each , let . Fix , by the previous lemma,

Corollary

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