Last Updated：2025/12/30
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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.
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