Subadditive set function

Everyday example of sigma sub-additivity: when sand is mixed with water, the bulk volume

Subadditive set function is a concept in the field of mathematics, particularly within the areas of measure theory and functional analysis. A set function, in this context, is a function that assigns a real number to each set in a given collection of sets, often subsets of a given set. The property of subadditivity is a crucial aspect in understanding the behavior of these functions, especially in applications related to probability theory, optimization, and economics.

Definition

A set function \( \nu: \mathcal{P}(X) \rightarrow \mathbb{R} \) is said to be subadditive if for any two subsets \( A \) and \( B \) of \( X \), the following inequality holds: \[ \nu(A \cup B) \leq \nu(A) + \nu(B) \] where \( \mathcal{P}(X) \) denotes the power set of \( X \), or the set of all subsets of \( X \), and \( \mathbb{R} \) is the set of real numbers. This definition can be extended to countable collections of sets, where the function is subadditive if for any countable collection of sets \( \{A_i\}_{i=1}^{\infty} \), it satisfies: \[ \nu\left(\bigcup_{i=1}^{\infty} A_i\right) \leq \sum_{i=1}^{\infty} \nu(A_i) \]

Examples

1. Lebesgue Measure: The Lebesgue measure on \( \mathbb{R}^n \) is a classic example of a subadditive set function. It assigns to each measurable set a non-negative real number (its "volume") and satisfies the subadditivity property among other properties.

2. Outer Measure: In measure theory, an outer measure is defined on all subsets of a given set and is always subadditive. It is designed to extend the concept of measure to sets that are not necessarily measurable in the traditional sense.

Properties

Subadditive set functions possess several important properties that make them useful in various mathematical analyses: - Monotonicity: If \( A \subseteq B \), then \( \nu(A) \leq \nu(B) \). This property follows directly from the definition of subadditivity. - Continuity from above: If \( \{A_i\} \) is a decreasing sequence of sets (i.e., \( A_{i+1} \subseteq A_i \) for all \( i \)), and \( \bigcap_{i=1}^{\infty} A_i = \emptyset \), then \( \lim_{i \to \infty} \nu(A_i) = 0 \), under certain conditions.

Applications

Subadditive set functions are utilized in various fields: - In probability theory, the concept of subadditivity is used to establish bounds on the probability of unions of events. - In optimization and operations research, subadditive functions help in formulating and solving problems related to resource allocation and cost minimization. - In economics, subadditivity can be applied to analyze economies of scale and to justify natural monopolies.

See Also

- Additive Set Function - Measure - Sigma-algebra - Monotone Class Theorem