Lebesgue integration: Encyclopedia II - Lebesgue integration - Construction of the Lebesgue integral

Lebesgue integration - Construction of the Lebesgue integral

The discussion that follows parallels the most common expository approach to the Lebesgue integral. In this approach the theory of integration has two distinct parts:

1. A theory of measurable sets and measures on these sets.
2. A theory of measurable functions and integrals on these functions.

Lebesgue integration - Measure theory

Measure theory initially was created to provide a detailed analysis of the notion of length of subsets of the real line and more generally area and volume of subsets of Euclidean spaces. In particular, it provided a systematic answer to the question of which subsets of R have a length. As was shown by later developments in set theory (see non-measurable set), it is actually impossible to assign a length to all subsets of R in a way which preserves some natural additivity and translation invariance properties. This suggests that picking out a suitable class of measurable subsets is an essential prerequisite.

Of course, the Riemann integral uses the notion of length implicitly. Indeed, the element of calculation for the Riemann integral is the rectangle [a, b] × [c, d], whose area is calculated to be (b-a)(d-c). The quantity b-a is the length of the base of the rectangle and d-c is the height of the rectangle. Riemann could only use planar rectangles to approximate the area under the curve because there was no adequate theory for measuring more general sets.

In the development of the theory in most modern textbooks (after 1950), the approach to measure and integration is axiomatic. This means that a measure is any function μ defined on certain subsets X of a set E which satisfies a certain list of properties. These properties can be shown to hold in many different cases.

The theory of measurable sets and measure (including definition and construction of such measures) is discussed in other articles. See measure.

Lebesgue integration - Integration

We will work in the following abstract setup: μ is a (non-negative) measure on a sigma-algebra X of subsets of E. For example, E can be Euclidean n-space Rⁿ or some Lebesgue measurable subset of it, X will be the sigma-algebra of all Lebesgue measurable subsets of E, and μ will be the Lebesgue measure. In the mathematical theory of probability μ will be a probability measure on a probability space E.

In Lebesgue's theory, integrals are limited to a class of functions called measurable functions. A function f is measurable if the pre-image of any closed interval is in X:

It can be shown that this is equivalent to requiring that the pre-image of any Borel subset of R be in X. We will make this assumption from now on. The set of measurable functions are closed under algebraic operations, but more importantly the class is closed under various kinds of pointwise sequential limits:

are measurable if the original sequence {f_k}, where k ∈ N, consists of measurable functions.

We build up an integral

for measurable complex-valued functions f defined on E in stages:

Indicator functions: To assign a value to the integral of the indicator function of a measurable set S consistent with the given measure μ, the only reasonable choice is to set:

Simple functions: We extend by linearity to the linear span of indicator functions:

where the sum is finite and the coefficients a_k are real numbers. Such a finite linear combination of indicator functions is called a simple function. Even if a simple function can be written in many ways as a linear combination of indicator functions, the integral will always be the same.

Non-negative functions: Let f be a non-negative measurable function on E which we allow to attain the value +∞, in other words, f takes values in the extended real number line. We define

We need to show this integral coincides with the preceding one, defined on the set of simple functions. There is also the question of whether this corresponds in any way to a Riemann notion of integration. It is not hard to prove that the answer to both questions is yes.

We have defined the integral of f for any non-negative extended real-values measurable function on E. For some functions ∫f will be infinite.

Signed functions: To handle signed functions, we need a few more definitions. If f is a function of the measurable set E to the reals (including ± ∞), then we can write

where

Note that both f⁺ and f⁻ are non-negative functions. Also note that

If

then f is called Lebesgue integrable. In this case, both integrals satisfy

and it makes sense to define

It turns out that this definition gives the desirable properties of the integral.

Complex valued functions can be similarly integrated, by considering the real part and the imaginary part separately.

Lebesgue integration - Intuitive interpretation

To get some intuition about the different approaches to integration, let us imagine that it is desired to find a mountain's volume (above sea level) and that the mountain's boundaries are marked out clearly (they are the boundaries of integration).

The Riemann-Darboux approach: Cut the mountain up into vertical slices, each one having a square base at sea-level. Pick a pair of points inside this square, one where elevation is highest and one where elevation is lowest. Associated to these two elevations are an upper volume and lower volume obtained by multiplying the elevations by the area of the square. The upper Riemann sum is the sum of the upper volumes of all the slices, and similarly for the lower Riemann sum. The Riemann integral exists if the upper and lower Riemann sums converge as the thickness of the slices decreases to 0.

The Lebesgue approach: Draw a contour map of the mountain. For each contour (or set of contours) with the lowest height, find the total area enclosed (within the map) by this set of contours, i.e. find the "measure" of this set of contours. Multiply this measure by the height represented by the set of contours: the product will be one summand of a "Lebesgue sum".

Then find a contour, or set of contours, which are at one step higher in height (lowest height of the remaining contours). Calculate the measure of the area enclosed by them. Multiply the measure by the difference in height (from the previous step), and the product will be another summand of the "Lebesgue sum".

Repeat this process for successive higher levels of contours, until the highest set of contours has been processed. The resulting sum is the linear span: each contour corresponding to an indicator function.

The sum can be refined by adding intermediate contours to the map: halving the difference between successive heights, then recomputing the sum. The Lebesgue integral is the limit of this process.

Lebesgue integration - Example

Consider the indicator function of the rational numbers, 1_Q. It is known 1_Q is nowhere continuous.

• 1_Q is not Riemann-integrable on [0,1]: No matter how the set [0,1] is partitioned into subintervals, each partition will contain at least one rational and at least one irrational number, since rationals and irrationals are both dense in the reals. Thus the upper Darboux sums will all be one, and the lower Darboux sums will all be zero.

• 1_Q is Lebesgue-integrable on [0,1]: Indeed it is the indicator function of the rationals so by definition

Adapted from the Wikipedia article "Construction of the Lebesgue integral", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki