nLab sigma-ideal

Idea

A $\sigma$-ideal is a collection of sets (either subsets of an ambient set or pure sets) that are considered ‘small’ in some fashion. Unlike the notion of ‘small’ in small category, this is not expected to be closed under most infinitary operations, but it is expected to be closed under countably infinitary operations, in particular under countable union.

If we use ‘large’ sets instead, then we have a $\delta$-filter.

Definitions

Let $X$ be a set. Then a $\sigma$-ideal on $X$ is a collection $\mathcal{I}$ of subsets of $X$ such that: 1. If $A \subset B$ and $B \in \mathcal{I}$, then $A \in \mathcal{I}$; 2. If $A_1, A_2, \ldots \in \mathcal{I}$, then there exists $B$ such that $B \in \mathcal{I}$ and $\bigcup_i A_i \subseteq B$; in light of (1), $B$ may be assumed to be the union $\bigcup_i A_i$ itself. 3. Some set belongs to $\mathcal{I}$; in light of (1), the empty set $\empty \in \mathcal{I}$.

A base of a $\sigma$-ideal is any collection satisfying (2,3); a base is precisely what generates a $\sigma$-ideal by closing under subsets. A subbase of a $\sigma$-ideal is any collection at all; a subbase generates a base by closing under countable unions.

Instead of a set, $X$ may easily be a proper class; then the elements of $\mathcal{I}$ may be restricted to subclasses that are actually sets. One may take $X$ to be the class of all pure sets; from the perspective of material set theory, this actually includes the general case above.

As defined above, a $\sigma$-ideal $\mathcal{I}$ is a subset of the power set (or power class) $\mathcal{P}X$; we can just as easily make $\mathcal{I}$ a subset of any complete lattice $\mathcal{L}$. Actually, it works just as well if $\mathcal{L}$ if replaced by a sup-semilattice with all countablary suprema, or probably even more generally (see ideal for some idea of how to do that). Note the grammar: a $\sigma$-ideal in $L$ but on $X$ (which is the same as in $\mathcal{P}X$).

Dually, a $\delta$-filter on $X$ is a collection $\mathcal{F}$ of subsets of $X$ such that: 1. If $A \subset B$ and $A \in \mathcal{F}$, then $B \in \mathcal{F}$; 2. If $A_1, A_2, \ldots \in \mathcal{F}$, then there exists $B$ such that $B \in \mathcal{F}$ and $B \subseteq \bigcap_i A_i$; in light of (1), $B$ may be assumed to be the intersection $\bigcap_i A_i$ itself. 3. Some set belongs to $\mathcal{F}$; in light of (1), the improper subset $X \in \mathcal{F}$.

Using de Morgan duality, $\delta$-filters and $\sigma$-ideals are essentially the same; we have

$$\mathcal{F} = \{ A \subseteq X \;|\; \neg{A} \in \mathcal{I} \}$$

and vice versa. In constructive mathematics, however, they are not equivalent. Also, the two notions are not equivalent when $X$ is a proper class.

Examples

• The power set of $X$ is both a $\sigma$-ideal and a $\delta$-filter on $X$; it is the improper $\sigma$-ideal on $X$. (Compare proper ideal.)
• The null sets in a measure space $X$ form a $\sigma$-ideal on $X$, while the full sets form the corresponding $\delta$-filter.
• The meagre sets in a topological space $X$ form a $\sigma$-ideal on $X$, while the comeagre sets form the corresponding $\delta$-filter.
• The pure sets form a $\sigma$-ideal on (or, equivalently in this case, in) the class of all pure sets; actually, this collection is closed under arbitrary unions rather than merely countablary unions, so we may call it a complete ideal.

Properties

Of course, any $\sigma$-ideal in an ideal. A $\sigma$-ideal is a $\sigma$-ring in its own right. In fact, a $\sigma$-ideal on $X$ is precisely simultaneously an ideal in and a sub-$\sigma$-ring of $\mathcal{P}X$. Dual results hold for $\delta$-filters.