2026-02-23 06:23:15.1771827795
I need help with a task.
Let ${ X_1 }$${ X_2 }$ sets and ${\mathscr{A_i} \subset \mathcal{P}(X_i)}$ two algebras on ${X_i}$ for i=1,2. Let also ${ \mu_i: \mathscr{A_i} \to [0, \infty ] }$ be 2 contents on ${X_i}$ and ${ \mathscr{A}=\mathscr{A_1}\times\mathscr{A_2}=\{C \subset X \vert \bigcup_{i=1}^{n}{A_i} \times {B_i}}$ with ${{A_i} \in \mathscr{A_1},{B_i} \in \mathscr{A_2}}$ and ${ n \in \mathbb{N}\} }$. Prove that there is exactly one content ${ \mu:\mathscr{A} \to [0, \infty ] }$ which holds that $${ \mu(A \times B)=\mu_1(A) \cdot \mu_2(B) }$$ for every ${{A_i} \in \mathscr{A_1}}$ and ${{B_i} \in \mathscr{A_2}}$.
In another thread someone suggested the monotone class theorem but I am not sure whether I can use it here (and I'm pretty sure it is not in my script anyway) and I tried to show it by using half-open intervals(Cuboid, not sure what you call it in english) but that got way to tedious and complicated.
Show that ${ \mu(A \times B)=\mu_1(A) \cdot \mu_2(B) }$
54 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtI need help with a task.
Let ${ X_1 }$${ X_2 }$ sets and ${\mathscr{A_i} \subset \mathcal{P}(X_i)}$ two algebras on ${X_i}$ for i=1,2. Let also ${ \mu_i: \mathscr{A_i} \to [0, \infty ] }$ be 2 contents on ${X_i}$ and ${ \mathscr{A}=\mathscr{A_1}\times\mathscr{A_2}=\{C \subset X \vert \bigcup_{i=1}^{n}{A_i} \times {B_i}}$ with ${{A_i} \in \mathscr{A_1},{B_i} \in \mathscr{A_2}}$ and ${ n \in \mathbb{N}\} }$. Prove that there is exactly one content ${ \mu:\mathscr{A} \to [0, \infty ] }$ which holds that $${ \mu(A \times B)=\mu_1(A) \cdot \mu_2(B) }$$ for every ${{A_i} \in \mathscr{A_1}}$ and ${{B_i} \in \mathscr{A_2}}$.
In another thread someone suggested the monotone class theorem but I am not sure whether I can use it here (and I'm pretty sure it is not in my script anyway) and I tried to show it by using half-open intervals(Cuboid, not sure what you call it in english) but that got way to tedious and complicated.