Guys can anyone help me with this question?
On a probability space let be filtration $F = (F_n)_{n \in N_0}$ and a real valued adaptive stochastic process $(X_n)_{n \in N_0}$ for all the Borelsets $ A \in B(R) $ we have $P[X_{n+1} \in A | F_n ] = P [X_{n+1} \in A ]$ P-almost surely
I must prove that the family $(X_n)_{n \in N_0}$ with respectt to $P$ is independent. I have no idea how to solve it. I will be thankful for any help
To prove family $\{X_n\}_{n \in \mathbb N}$ is independent, it is sufficient and necessary (via definition) to prove that family $\{X_n\}_{ n \in \{1,...,k\}}$ is independent (for any $k \in \mathbb N$). Hence take any $k$ and borel sets $A_1,...,A_k$. We need to show $$ \mathbb P(\bigcap_{j=1}^k \{X_j \in A_j\}) = \prod_{j=1}^k \mathbb P(X_j \in A_j)$$
How can we use our assumption? We're start with left side and condition on $F_{k-1}$, getting:
$$ \mathbb P(\bigcap_{j=1}^k \{X_j \in A_j\}) = \mathbb E \mathbb E[ \prod_{j=1}^k 1_{ \{X_j \in A_j\}}| F_{k-1}] = \mathbb E[\prod_{j=1}^{k-1}1_{\{X_j \in A_j\}} \mathbb E[ 1_{\{X_k \in A_k\}} | F_{k-1}]] $$ Now use our assumption getting $\mathbb P(X_k \in A_k)$ in the middle which can be throw out of expectation (since it's a constant), getting: $$\mathbb P(\bigcap_{j=1}^k \{X_j \in A_j\})= \mathbb P(X_k \in A_k) \mathbb E[ \prod_{j=1}^{k-1} 1_{\{X_j \in A_j\}}] = \mathbb P(X_k \in A_k) \mathbb P(\bigcap_{j=1}^{k-1} \{X_j \in A_j\})$$
Can you proceed inductivelly now? You need to condition the rest by $F_{k-2}$ then what's left by $F_{k-3}$ and so on, everytime using your assumption, at last getting what we need to prove, that is $$ \mathbb P( \bigcap_{j=1}^k \{X_j \in A_j\}) = \prod_{j=1}^k \mathbb P(X_j \in A_j)$$
