Atom (measure theory)

In measure theory, an atom (or sometimes called an indivisible set) with respect to a measure space (X, Σ, μ) is a measurable set A ∈ Σ such that μ(A) > 0, and for any measurable subset B ⊆ A, either μ(B) = 0 or μ(B) = μ(A). In other words, an atom is a set of positive measure that cannot be further subdivided into sets of smaller positive measure.

Formal Definition:

Let (X, Σ, μ) be a measure space. A set A ∈ Σ is called an atom if:

1. μ(A) > 0
2. For all B ∈ Σ, if B ⊆ A, then either μ(B) = 0 or μ(B) = μ(A).

Properties and Implications:

• Indivisibility: The defining property of an atom is its indivisibility in terms of measure. Any measurable subset of an atom has either zero measure or the same measure as the atom itself.

• Atomic Measures: A measure space is called atomic if X can be written as a countable union of atoms and a set of measure zero. Measures that are atomic are often called purely atomic measures.

• Non-Atomic Measures: A measure space is called non-atomic (or atomless) if it contains no atoms. In a non-atomic measure space, any set of positive measure can be further divided into sets of strictly smaller positive measure. The Lebesgue measure on the real line is a common example of a non-atomic measure.

• Decomposition Theorem: Measures can be decomposed into an atomic part and a non-atomic part. Specifically, every measure can be uniquely expressed as the sum of a purely atomic measure and a non-atomic measure. This decomposition is crucial in understanding the structure of general measures.

• Relationship to Point Masses: In probability theory, an atom corresponds to a point where a probability mass is concentrated. For instance, in a discrete probability distribution, each point with a positive probability mass forms an atom with respect to the probability measure.

Examples:

• Discrete Measure: Consider a discrete measure on the integers, where μ({n}) = 1 for all integers n. Each singleton set {n} is an atom with measure 1.

• Dirac Delta Measure: The Dirac delta measure δ_x concentrated at a point x is defined such that δ_x(E) = 1 if x ∈ E and δ_x(E) = 0 if x ∉ E. The set {x} is an atom with measure 1.

• Lebesgue Measure: The Lebesgue measure on the real line is non-atomic. For any set A with positive Lebesgue measure, one can always find a subset B of A with strictly smaller positive Lebesgue measure.

Significance:

Atoms play a significant role in measure theory and probability. They provide a way to characterize the nature of measures and distributions. Understanding the atomic and non-atomic components of a measure helps in analyzing its properties and behavior. The presence or absence of atoms greatly influences various theoretical results and practical applications.