1.3 Lebesgue measure and its properties

Lebesgue measure is a cornerstone of measure theory, extending the concept of length to more complex sets. It assigns non-negative real numbers to subsets of Euclidean space, starting with intervals and expanding to higher dimensions.

Key properties include translation invariance, regularity, and countable additivity. Lebesgue measure is more comprehensive than Borel measure and relates closely to Hausdorff measure, making it essential for understanding measure theory's foundations.

Lebesgue Measure on the Real Line

Definition and Construction

• The Lebesgue measure assigns a non-negative real number to subsets of n-dimensional Euclidean space
  • Generalizes the notion of length, area, and volume
• Construction begins by defining the measure for intervals and extending it to more complex sets using outer measure
  • Lebesgue measure of an interval $[a, b]$ is defined as $b - a$ (consistent with the notion of length)
  • For a set $A$ on the real line, the Lebesgue outer measure is the infimum of the sum of lengths of all countable collections of open intervals that cover $A$
• A set $A$ is Lebesgue measurable if for every set $X$, the outer measure of $X$ equals the sum of outer measures of $A \cap X$ and $A^c \cap X$ ($A^c$ denotes the complement of $A$)

Extension to Higher Dimensions

• Lebesgue measure can be extended to higher dimensions using product measures
  • For a rectangle $R$ in $\mathbb{R}^n$, the Lebesgue measure is the product of the lengths of its sides
  • Lebesgue outer measure and measurability in $\mathbb{R}^n$ are defined similarly to the one-dimensional case
    • Uses countable collections of open rectangles instead of open intervals
• Enables the measurement of volume in higher-dimensional spaces ($\mathbb{R}^2$, $\mathbb{R}^3$, etc.)
• Provides a consistent way to measure sets across different dimensions

Properties of Lebesgue Measure

Translation Invariance and Regularity

• Translation invariance: For any Lebesgue measurable set $A$ and real number $t$, the translation $A + t$ is also Lebesgue measurable
  • Measure of $A + t$ is equal to the measure of $A$
  • Ensures that the measure is independent of the set's position on the real line
• Regularity: For any Lebesgue measurable set $A$, its measure equals:
  • Infimum of measures of all open sets containing $A$
  • Supremum of measures of all compact sets contained in $A$
  • Allows approximation of the measure using open and compact sets

Countable Additivity and Completeness

• Countable additivity: For a countable collection of pairwise disjoint Lebesgue measurable sets $\{A_n\}$, the measure of their union equals the sum of individual measures
  • Mathematically, $m(\bigcup_{n=1}^{\infty} A_n) = \sum_{n=1}^{\infty} m(A_n)$
  • Ensures consistency when measuring disjoint sets
• Completeness: Every subset of a Lebesgue measurable set with measure zero is also Lebesgue measurable
  • Extends measurability to "small" subsets of measure zero sets
• Borel-Cantelli lemma: For a countable collection of Lebesgue measurable sets $\{A_n\}$ with a finite sum of measures, the set of points belonging to infinitely many $A_n$ has Lebesgue measure zero
  • Useful in probability theory and ergodic theory

Lebesgue Measure vs Other Measures

Comparison with Borel Measure

• Borel measure is defined on the Borel $\sigma$-algebra (smallest $\sigma$-algebra containing all open sets in a topological space)
• Lebesgue measure extends the Borel measure
  • Every Borel measurable set is Lebesgue measurable, but not vice versa
  • Vitali set is an example of a Lebesgue measurable set that is not Borel measurable
• Lebesgue measure is more comprehensive than Borel measure

Comparison with Hausdorff Measure

• Hausdorff measure generalizes Lebesgue measure by assigning a measure to sets according to their Hausdorff dimension
  • Extends the notion of dimension to non-integer values
  • Useful for measuring the "size" of fractal sets (Cantor set, Koch snowflake)
• For sets with integer Hausdorff dimension $n$, the $n$-dimensional Hausdorff measure coincides with the $n$-dimensional Lebesgue measure up to a constant factor
• Both Lebesgue and Hausdorff measures are translation invariant and countably additive
  • Hausdorff measure is not regular, unlike Lebesgue measure
• Lebesgue measure is more suitable for "smooth" sets, while Hausdorff measure is more appropriate for "irregular" or fractal sets