In real analysis, we often read about sets, the real line, and intervals on the real line. However, there exists a special measure called the Lebesgue measure on the real line. This can be defined with the help of intervals defined on the real number line. In this article, you will learn about the Lebesgue measure on the real number line, how to define the Lebesgue measure for a given set or interval, along with suitable examples.

Lebesgue Measure Definition

The Lebesgue measure is the standard method of allocating a measure to subsets of n-dimensional Euclidean space. It is employed throughout the real analysis and very particularly to determine Lebesgue integration. For n = 1, 2, or 3, the Lebesgue measure overlaps with the regular length, area, or volume measurement. Here, the volume can be termed as n-dimensional volume or n-volume.

Lebesgue Measurable Set

Sets that can be allotted a Lebesgue measure are known as Lebesgue-measurable sets, and the Lebesgue-measurable set A is denoted by λ(A).

For any interval, I = [a, b] in the set of real numbers R, and let l(I) = b – a be the length of the interval defined.

For any subset of R, say E ⊆ R, the Lebesgue outer measure λ*(E) is defined as an infimum, which is represented by:

Here, some sets E satisfies a specific criterion called Caratheodory criterion such that it requires for every A ⊆ R.

Therefore, λ*(A) = λ*(A ⋂ E) + λ*(A ⋂ E^C)

In simple terms, we can say that a set E ⊂ R is called Lebesgue measurable if for every subset A of R, λ*(A) = λ*(A ∩ E) + λ*(A ∩ E^c)

Alternatively, we can get the Lebesgue measurable set as follows:

Suppose a set E is a Lebesgue measurable set, then the Lebesgue measure of E is said to be its outer measure λ*(E), and it can be expressed as λ(E).

Lebesgue Measurable Set Examples

Some examples of Lebesgue measurable sets are listed below:

• Suppose a and b are any two real numbers such that the closed interval [a, b] is Lebesgue measurable, and its length is given by b − a.
• Similarly, an open interval (a, b) contains the same Lebesgue measure, i.e. b – a, since the difference between the two sets consists only of the endpoints, i.e. a and b, and holds a zero measure.
• Consider two intervals, [a, b] and [c, d]. Here, the Cartesian product of these two intervals is called Lebesgue-measurable, and its Lebesgue measure is given by the product (b – a) and (d – c), i.e. (b − a)(d − c). This represents the area of the rectangle formed from these points.
• The Lebesgue measure is equal to 0 for any countable set of real numbers. For instance, for a set of algebraic numbers, the Lebesgue measure is 0, even if this set is said to be dense in R.
• Some examples of uncountable sets containing Lebesgue measure 0 include the Cantor set, the set of Liouville numbers, and so on.

Also, check: Intervals as Subsets of R

Properties of Lebesgue Measure on the Real Line

Below are a few important properties of the Lebesgue measure of real numbers.

Consider a set of real numbers E, and if its Lebesgue measure is denoted by λ(E) (if it is defined), then it holds the following properties.

Extends length

For every interval I, λ(I) = l(I), which means Lebesgue measure of I = length of the interval I.

Monotone

If A is a subset of B and B is a subset of R (a set of real numbers), i.e. A ⊂ B ⊂ R then we can write the expression 0 ≤ λ(A) ≤ λ(B) ≤ ∞.

Translation invariant

For every subset A of the set of real numbers R and for every point x₀ ∈ R, we can define the set A as:

A + x₀ = {x + x₀: x ∈ A}

Also, λ(A + x₀) = λ(A).

Countable additive

Suppose A and B are two disjoint subsets of the set of real numbers R, then λ(A U B) = λ(A) + λ(B).

Also, if {Ai} represents the sequence of disjoint sets, then the Lebesgue measure of this sequence is given by