Inner Measure Is Sup Of Closed Sets

Inner Measure: The Supremum of Closed Sets

This article delves into the concept of inner measure, specifically focusing on its definition as the supremum of the measures of closed sets contained within a given set. We'll explore its properties, contrast it with outer measure, and discuss its significance in measure theory. Understanding inner measure is crucial for grasping more advanced topics like measurable sets and the Lebesgue measure.

What is Inner Measure?

The inner measure of a set A, denoted as m(A)*, is defined as the supremum (least upper bound) of the measures of all closed sets contained within A. In simpler terms, we look at all the closed sets that fit entirely inside A, measure each of them, and then find the largest possible measure we can achieve. This provides a measure from the "inside" looking out, contrasting with the outer measure approach.

Formal Definition:

Let (X, Σ, μ) be a measure space. For any set A ⊂ X, the inner measure of A is defined as:

m(A) = sup{μ(F) : F ⊂ A, F is closed}*.

Relationship to Outer Measure:

The inner measure is a concept intrinsically linked to the outer measure, m(A). Outer measure, conversely, is defined as the infimum (greatest lower bound) of the measures of open sets containing A. The outer measure approaches the set from the "outside," covering it with open sets.

The relationship between inner and outer measure is fundamental: A set A is measurable if and only if its inner and outer measures are equal, i.e., m(A) = m*(A). This equality is a key criterion for measurability in measure theory.

Properties of Inner Measure:

Inner measure possesses several important properties, many mirroring (though not always identically) those of outer measure:

• Monotonicity: If A ⊂ B, then m(A) ≤ m(B). This is intuitive; a smaller set cannot have a larger inner measure.

• Non-negativity: m(A) ≥ 0 for all A. Measures are always non-negative.

• Subadditivity (for countable unions): For a countable collection of sets {A_i}, m(∪_i A_i) ≥ Σ_i m(A_i). This isn't necessarily an equality, unlike outer measure.

• Inner Measure of Closed Sets: For any closed set F, m(F) = μ(F). The inner measure of a closed set is simply its measure. This directly stems from the definition.

Significance and Applications:

The inner measure provides a valuable tool for characterizing measurable sets. It helps to establish the necessary and sufficient conditions for a set to be measurable within a given measure space. Understanding the interplay between inner and outer measure is crucial for grasping the complexities of integration theory and advanced probability theory. Furthermore, the concept finds applications in various fields, including:

• Lebesgue Integration: Inner measure plays a key role in developing the theory of Lebesgue integration, a powerful generalization of the Riemann integral.

• Probability Theory: In probability, the inner measure helps to define and analyze probabilities of events, especially in more complex probability spaces.

• Fractal Geometry: While not directly used in the core definition of fractal dimensions, the concepts related to inner measure inform the analysis of irregular sets and their measure properties.

Conclusion:

The inner measure, defined as the supremum of measures of closed subsets, is a fundamental concept in measure theory. Its relationship with outer measure is central to determining measurability, and it plays a crucial role in developing sophisticated mathematical tools used across various branches of mathematics and its applications. A deep understanding of inner measure enhances one’s comprehension of advanced measure theory and its practical applications.