Lebesgue Measure

The concept of Lebesgue measure is central to real analysis and serves as a foundation for modern measure theory, providing a way to rigorously define the measure of subsets of $n$-dimensional Euclidean space ${\mathbb{R}}^n$. Developed by Henri Lebesgue in the early 20th century, it generalizes the notion of the lengths of intervals, areas of surfaces, and volumes of solids in a way that is particularly useful for the integration of functions with respect to this measure, leading to the Lebesgue integral.

Definition

The Lebesgue measure, denoted as $\lambda$, is a measure on the $\sigma$-algebra of Lebesgue measurable sets in ${\mathbb{R}}^n$ that satisfies the following properties:

1. Non-negativity: For every Lebesgue measurable set $E\subset {\mathbb{R}}^n$, $\lambda (E)\ge 0$.
2. Null empty set: $\lambda (\varnothing )=0$.
3. Countable additivity (or $\sigma$-additivity): If $\{E_i{\}}_{i=1}^{\infty }$ is a countable collection of pairwise disjoint Lebesgue measurable sets, then $\lambda \left({\bigcup }_{i=1}^{\infty }{E}_i\right)={\sum }_{i=1}^{\infty }\lambda (E_i)$.
4. Translation invariance: For any Lebesgue measurable set $E$ and any vector $v\in {\mathbb{R}}^n$, $\lambda (E+v)=\lambda (E)$, where $E+v=\{x+v:x\in E\}$.

Construction

The construction of the Lebesgue measure can be briefly described in several steps:

1. Defining measure for intervals: Start by defining the measure of any box (or rectangle) in ${\mathbb{R}}^n$ as the product of its side lengths. For a box $B=[a_1,b_1]\times [a_2,b_2]\times \cdots \times [a_n,b_n]$, its Lebesgue measure is $\lambda (B)={\prod }_{i=1}^n(b_i-a_i)$.

2. Outer measure: For any subset $A\subset {\mathbb{R}}^n$, define its outer measure, ${\lambda }^{\ast }(A)$, as the infimum of the sums of the measures of collections of boxes that cover $A$.

3. Measurable sets: A set $E\subset {\mathbb{R}}^n$ is Lebesgue measurable if for every subset $A\subset {\mathbb{R}}^n$, ${\lambda }^{\ast }(A)={\lambda }^{\ast }(A\cap E)+{\lambda }^{\ast }(A\cap E^c)$, where $E^c$ denotes the complement of $E$ in ${\mathbb{R}}^n$.

4. Measure of measurable sets: The Lebesgue measure of a Lebesgue measurable set $E$ is simply $\lambda (E)={\lambda }^{\ast }(E)$.

Significance and Applications

The significance of the Lebesgue measure lies in its ability to measure a broader class of sets and integrate a wider class of functions than possible with the classical Riemann integral. This generality makes it invaluable in various fields of mathematics and applied sciences, including probability theory, functional analysis, and mathematical physics.

One of the key applications of the Lebesgue measure is in defining the Lebesgue integral, which allows for the integration of functions that are not Riemann integrable and provides powerful convergence theorems like the Dominated Convergence Theorem and the Monotone Convergence Theorem.

Mathematical Formalisms

The mathematical elegance of Lebesgue’s approach lies in its use of outer measures and measurable sets. For a function $f:{\mathbb{R}}^n\to \mathbb{R}$, the Lebesgue integral $\int fd\lambda$ is defined by approximating $f$ from below and above with simple functions (functions that take a finite number of values) whose integrals can be easily computed using the measure of the sets on which these values are taken.

Conclusion

The Lebesgue measure is a cornerstone of modern analysis, offering a robust framework for measuring sets and integrating functions over these sets. Its development not only solved longstanding problems in mathematics but also opened up new avenues for research and application in analysis and beyond.