Independence (2). Let \((\Omega, \mathscr{A}, P)\) be a probability space and assume that the \(\sigma\)-algebras \(\mathscr{B}, \mathscr{C} \subset \mathscr{A}\) are independent (see Problem 5.11). Show that \(u \in \mathcal{M}^{+}(\mathscr{B})\) and \(w \in \mathcal{M}^{+}(\mathscr{C})\) satisfy

\[\int u w d P=\int u d P \cdot \int w d P\]

and that for \(u \in \mathcal{M}(\mathscr{B})\) and \(w \in \mathcal{M}(\mathscr{C})\)

\[u w \in \mathcal{L}^{1}(\mathscr{A}) \Longleftrightarrow u \in \mathcal{L}^{1}(\mathscr{B}) \quad \text { and } \quad w \in \mathcal{L}^{1}(\mathscr{C})\]

Find an example proving that this fails if \(\mathscr{B}\) and \(\mathscr{C}\) are not independent.

[start with simple functions and use Beppo Levi's theorem.]

Data from problem 5.11

Transcribed Image Text:

Independence (1). Let (2, 4,P) be a probability space and let B, CCA be two sub-o- algebras of. We call and independent, if P(BC)=P(B)P(C) VBEB, CEC. Assume now that = o(G) and C = o(H), where 9, H are n-stable collections of sets. Prove that and are independent if, and only if, P(Gn H)=P(G)P(H) GE9, HEH.