sigmaideal
Sigma-ideal, or sigma-ideal, is a concept from set theory and measure theory describing a special collection of subsets of a given set X. A sigma-ideal I on X is a nonempty family of subsets of X that is downward closed (if A is in I and B is a subset of A, then B is in I) and closed under countable unions (if A1, A2, A3, … are in I, then their union is in I). It is common to also require that I is proper, meaning I does not contain all subsets of X (i.e., I ≠ P(X)).
Key properties and notions include:
- I contains the empty set and is closed under taking subsets and under countable unions.
- The notion is designed so that I captures “small” or “negligible” sets with respect to a context,
- The sigma-ideal generated by a family F of subsets of X is the smallest sigma-ideal that contains
- On an infinite set X, the collection of all countable subsets of X is a sigma-ideal when
- In a topological space, the collection of meager (first category) sets forms a sigma-ideal.
- In a measure space (X, Σ, μ) with a complete measure, the collection of sets of μ-measure zero
- A sigma-ideal is not typically closed under complements, so it is not a sigma-algebra.
- The quotient Boolean algebra P(X)/I, formed by modulo-ing out by I, is a common object of
- The concept is central in descriptive set theory and in the study of “small” sets relative
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