Prof. Boris Tsirelson の「確率変数」

良心的な人だ。

http://www.tau.ac.il/~tsirel/Courses/MeasContin/lect2.pdf より引用：

A remark on terminology. Some authors call a measurable map φ from a probability space to a measurable space (Y, B) “random element of Y ” with special cases like “random vector”, “random sequence”, “random function” etc. That is nice, but two objections arise. First, no one says “random real number” instead of “random variable”! Second, some people insist that, say, a “random vector” must be a single vector chosen at random, or somehow typical, but surely not a map!

Other authors call such φ “Y-valued random variable” which avoids the two objections mentioned above but leads to cumbersome phrases like “R^T-valued random variable” instead of “random function”. Also, one complains that a so-called random variable is neither random nor variable!

Another choice is, between functions and equivalence classes. I hesitate
but I must choose. . .

で、定義は次のようになる。

Definition. Let (X, A, µ) be a probability space and (Y, B) a measurable space.

(a) A random element of Y is an equivalence class (w.r.t. the equivalence relation “equal almost everywhere”) of measurable functions X → Y .

(b) The distribution of a random element φ is the measure ν_φ on (Y, B) defined by

• ν_φ(B) = µ(Φ(B)),

where Φ(B) is the equivalence class of {x : φ(x) ∈ B}.

(c) The measure subalgebra σ(φ)/∼^μ ⊂ A/∼^μ generated by φ is defined by

• σ(φ)/∼^μ = Φ(B) = {Φ(B) : B ∈ B}

(where Φ is as in (b)); the corresponding (in the sense of 2a11) σ-algebra
σ(φ) ⊂ A_μ (containing all null sets) is the σ-algebra generated by φ.

測度代数の定義は、

The set of all equivalence classes of measurable sets,

• A/∼^μ = A_μ/∼^μ ,

is called the measure algebra of (X, AA). It is closed under the finite and countable operations.

測度代数は完備ブール代数〈complete boolean algebra〉になる。位相空間としての2^X/〜^μに測度代数は埋め込めて、埋め込み先では閉集合になる。完備は、任意の部分集合が上界を持つこと。「任意に総和可能」と言ってもいい。