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つづき

https://en.wikipedia.org/wiki/Indexed_family
Indexed family

In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers is a collection of real numbers, where a given function selects one real number for each integer (possibly the same).

More formally, an indexed family is a mathematical function together with its domain I and image X. Often the elements of the set X are referred to as making up the family. In this view, indexed families are interpreted as collections of indexed elements instead of functions. The set I is called the index (set) of the family, and X is the indexed set. Sequences are one type of families with the specific domains.

Mathematical statement
Definition. Let I and X be sets and f a function such that
略

The symbol x is used to indicate that x_i is an element of X.), then this establishes an indexed family of elements in X indexed by I, which is denoted by (x_i)_{i∈ I} or simply (xi), when the index set is assumed to be known.

The index set I is not restricted to be countable, and a subset of a power set may be indexed, resulting in an indexed family of sets. Sequences are one type of families as a sequence is defined as a function with the specific domain (an interval of integers, the set of natural numbers, or the set of first n natural numbers, depending on what sequence is defined and what definition is used).

https://ja.wikipedia.org/wiki/%E7%8B%AC%E7%AB%8B_(%E7%A2%BA%E7%8E%87%E8%AB%96)
独立 (確率論)
一般に、（有限とは限らない）事象の族 Aλ が独立であるとは、その任意の有限部分族 Aλ1,Aλ2,・・・,Aλnに対して
P(Aλ1∩Aλ2∩・・・∩Aλn)=P(Aλ1)P(Aλ2)・・・P(Aλn)
が成立することをいう。
(引用終り)
以上