Lebesgue Integral | Research & Encyclopedia Articles

Lebesgue Integral

The Lebesgue integral is one of the most important and powerful tools in mathematical analysis. The Lebesgue integral can be defined in very general settings and the vector space of real or complex valued Lebesgue integrable functions can be organized into a Banach space. In order to define the Lebesgue integral of a function it is necessary to develop the most basic concepts of measure theory.

Let be a nonempty set. A collection A of subsets of is called a -algebra if it satisfies the following conditions:

• (1) both the empty set and the set are elements of A,
• (2) if A ⊆ belongs to A then its compliment ∖ A also belongs to A,
• (3) if A₁, A₂, ... is a countable collection of sets in A then both

The pair (, A) is called a measurable space. Here is an example of a measurable space. Let ℜ be the set of real numbers. Then let ℬ be the intersection of all -algebras of subsets of ℜ that contain the open intervals. There is at least one -algebra in ℜ that contains the open interval, namely, the -algebra of all subsets of ℜ. And it can be shown that the intersection of an arbitrary family of -algebras is again a -algebra. Thus the collection ℬ exists and is a -algebra of subsets of ℜ containing the open intervals. Because each open subset of ℜ is a countable union of open intervals, it follows that ℬ contains all the open subsets of ℜ. Since each closed subset of ℜ is the compliment of an open set, we also find that ℬ contains all the closed subsets of ℜ. The -algebra ℬ is called the -algebra of Borel sets in ℜ. Thus the pair (ℜ, ℬ) forms a measurable space.

A function :A [0,] is called a measure if it satisfies the conditions:

• (4) () = 0,
• (5) if A₁, A₂, ... is a countable collection of disjoint sets in A then

Notice that the values taken on by a measure may include the symbol and this must be treated appropriately. For example, in condition (5) it may happen that (A_n) < for each n = 1, 2, ... , that (A_n) = and then the sum on the right hand side of (5) must diverge to . The triple (, A, ) is called a measure space. It can be shown that there exists a unique measure defined on the -algebra ℬ of Borel sets in ℜ such that for each interval I ℜ the value of (I) is the length of the interval. In particular, (I) = if I is in an infinite interval. Of course is defined for all sets in ℬ, not just the intervals, in such a way that it satisfies the requirements of a measure. This particular measure on ℬ is called Lebesgue measure. Thus the triple (ℜ, ℬ, ) forms a measure space. There is also a larger -algebra ℒ in ℜ called the -algebra of Lebesgue measurable sets that contains ℬ as a sub--algebra. And the measure can be extended to a measure on ℒ so that (ℜ, ℒ, ) is a measure space.

A function ƒ: ℜ is said to be measurable with respect to the -algebra A if ƒ^-1(B) is contained in A for each Borel set B in ℬ. When there is only one -algebra being considered in the domain of such a function, we say more simply that the function is measurable. For example, let A be a set in A, that is, A is a measurable subset of . Then define a function x_A: ℜ by x_A(x) = 1 if x is a point in A, and x_A(x) = 0 if x not a point in A. The function x_A is called the characteristic function of A, and it is easy to see that x_A is a measurable function. More generally, if A₁, A₂, ... A_n is a sequence of measurable subsets of and c₁, c₂, ... c_N is a corresponding set of nonnegative real numbers, then the function ϕ: ℜ defined by

is a measurable function. Here ϕ is an example of a simple function, that is, a measurable function that takes on finitely many nonnegative values. It can be shown that every simple function can be written as a finite linear combination of characteristic functions of measurable sets in the same manner that we have defined ϕ.

We are now in position to define the Lebesgue integral. First of all, let A be a measurable subset of . Then the Lebesgue integral over the space of the characteristic function x_A of A is defined to be the measure of A. That is, we define

More generally, if ϕ is a simple function written as before, then the Lebesgue integral of ϕ over the space is defined by

Here it is important to recognize the convention that if c_n = 0 and (A_n) = for some integer n, then c_n (A_n) = 0. Next we suppose that ƒ: [0,) is a measurable function. In this case we define

where the supremum is taken over the set of all simple functions ϕ such that 0 ϕ(x) ƒ(x) at each point x in . Notice that the value of the integral of a nonnegative measurable function ƒ could be the symbol . This is convenient for establishing the basic properties of the integral.

Now assume that ƒ: ℜ is measurable. Define

ƒ⁺(x) = ½|ƒ(x)| + ½ƒ(x) and ƒ^-(x) = ½|ƒ(x)| - ½ ƒ(x)

so that ƒ⁺ and ƒ^- are nonnegative valued measurable functions,

|ƒ(x)| = ƒ⁺(x) + ƒ^-(x) and ƒ(x) = ƒ⁺(x) - ƒ^-(x).

We say that ƒ is integrable over the measure space (, A, ) if

If ƒ is integrable it follows that

Therefore, if ƒ is integrable we define the value of the Lebesgue integral of ƒ over the measure space (, A, ) by

Clearly the value of the integral of an integrable function is always a real number and never the symbol . If ƒ and g are both integrable, if and are real numbers, then ƒ + g is integrable and

Thus the Lebesgue integral is a linear transformation on the vector space of integrable functions. Here is an example of one of the important convergence theorems for the Lebesgue integral: suppose that ƒ₁(x), ƒ₂(x), ... is a sequence of measurable functions such that

for each point x in . Also assume that there exists a nonnegative valued integrable function g: [0, ) such that |ƒ_n(x)| g(x) for all x in and all positive integers n. Then each function ƒ_n is integrable, the function F: ℜ is measurable, F is integrable, and

We have already noted that (ℜ, ℒ, ) is an example of a measure space and therefore the concepts described here for the Lebesgue integral apply in this particular measure space. For example, let [u, v] ℜ be a closed interval and let ƒ: [u, v]: ℜ be a Riemann integrable function. It will be convenient to set ƒ(x) = 0 if x is a real number not in the interval [u, v]. Then it can be shown that ƒ is measurable with respect to the -algebra ℒ, ƒ is integrable over the measure space (ℜ, ℒ, ), and the value of the Lebesgue integral of ƒ on ℜ is equal to the value of the Riemann integral of ƒ on the interval [u, v]. On the other hand, there exist measurable functions g: ℜ ℜ, with g(x) = 0 for real numbers x not in [u,v], and such that g is Lebesgue integrable over (ℜ, ℒ, ) but the restriction of g to the interval [u, v] is not Riemann integrable. Thus the Lebesgue method of integration extends the Riemann method of integration to a wider class of functions. However, the Riemann method of integration is certainly not obsolete. Suppose that h: [u, v] ℜ is a continuous function, and therefore both Riemann and Lebesgue integrable. If we wish to find an approximate numerical value for the integral of h over [u, v] the principles used to define the Riemann integral of h may well be more useful.

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