direct product

If A and B are sets, their direct product is the set of ordered pairs (a,b) with a in A and b in B.

Theorem 3.2.3. A periodic Abelian group is the direct product of its Sylow subgroups, S(p).

A Boolean ring of order 2ⁿ (or dimension n) may be constructed as the direct product of n Boolean rings of dimension one.

Let us start with the definition of the direct product of two probability measures. Let P_1 and P_2 be probability measures on 𝛺₁ and 𝛺₂, respectively, and denote 𝛺₁⨯𝛺₂ by 𝛺. A probability measure P on 𝛺 with 𝔇(P)=𝔇(P_1)⨯𝔇(P_2) is called the direct product of P_1 and P_2 (written P_1⨯P_2) if

P(B_1⨯B_2)=P(B_1)P(B_2) B_i∈𝔇(P_i);i=1,2.

The probability space (𝛺,P) is called the direct product of (𝛺₁,P_1) and (𝛺₂,P_2), written

(𝛺,P)=(𝛺₁,P_1)⨯(𝛺₂,P_2).

For example, the Lebesgue measure on [0, 1]² is the direct product of that on [0, 1] and itself.