Show that for any finite collection $(\Omega_i, {\mathcal F}_i, \mu_i)_{i \in A}$ of probability spaces, there exists a unique probability measure $\prod_{i \in A} \mu_i$ on $(\prod_{i \in A} \Omega_i, \prod_{i \in A} {\mathcal F}_i)$ such that $$\displaystyle (\prod_{i \in A}\mu_i)(\prod_{i \in A} E_i) = \prod_{i \in A} \mu_i(E_i)$$ whenever $E_i \in {\mathcal F}_i$ for $i \in A$. Furthermore, show that $$\displaystyle \prod_{i \in A}\mu_i = (\prod_{i \in A_1}\mu_i) \times (\prod_{i \in A_2}\mu_i)$$ for any partition $A = A_1 \uplus A_2$ (after making the obvious identification between $\prod_{i \in A} \Omega_i$ and $(\prod_{i \in A_1} \Omega_i) \times (\prod_{i \in A_2} \Omega_i)$). Thus for instance one has the associativity property $$\displaystyle \mu_1 \times \mu_2 \times \mu_3 = (\mu_1 \times \mu_2) \times \mu_3 = \mu_1 \times (\mu_2 \times \mu_3)$$ for any probability spaces $(\Omega_i, {\mathcal F}_i, \mu_i)$ for $i=1,\dots,3$.

Homepage › Solution manuals › Terence Tao › Probability Theory › Exercise 2.3 (Finite products)

Exercise 2.3 (Finite products)

Show that for any finite collection ${(Ω_{i}, ℱ_{i}, μ_{i})}_{i \in A}$ of probability spaces, there exists a unique probability measure $\prod_{i \in A} μ_{i}$ on $(\prod_{i \in A} Ω_{i}, \prod_{i \in A} ℱ_{i})$ such that

(\prod_{i \in A} μ_{i}) (\prod_{i \in A} E_{i}) = \prod_{i \in A} μ_{i} (E_{i})

whenever $E_{i} \in ℱ_{i}$ for $i \in A$ . Furthermore, show that

\prod_{i \in A} μ_{i} = (\prod_{i \in A_{1}} μ_{i}) \times (\prod_{i \in A_{2}} μ_{i})

for any partition $A = A_{1} ⊎ A_{2}$ (after making the obvious identification between $\prod_{i \in A} Ω_{i}$ and $(\prod_{i \in A_{1}} Ω_{i}) \times (\prod_{i \in A_{2}} Ω_{i})$ ). Thus for instance one has the associativity property

μ_{1} \times μ_{2} \times μ_{3} = (μ_{1} \times μ_{2}) \times μ_{3} = μ_{1} \times (μ_{2} \times μ_{3})

for any probability spaces $(Ω_{i}, ℱ_{i}, μ_{i})$ for $i = 1, \dots, 3$ .

Exercise 2.3 (Finite products)

Add answer