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#95;1 and P#95;2 be probability measures on #92;Omega#95;1 and #92;Omega#95;2, respectively, and denote #92;Omega#95;1#92;times#92;Omega#95;2 by #92;Omega. A probability measure P on #92;Omega with #92;mathfrak#123;D#125;#92;left(P#92;right)#61;#92;mathfrak#123;D#125;#92;left(P#95;1#92;right)#92;times#92;mathfrak#123;D#125;#92;left(P#95;2#92;right) is called the direct product of P#95;1 and P#95;2 (written P#95;1#92;timesP#95;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.