I was looking at theorem 22.1 here , where one wants to show that a Brownian bridge $(\tilde B_t)_t$ is independent from the random variable $B(1)$. This seems a bit tricky to me: one is a process, the other one is a random variable.
What is done in the proof is just to take a partition $t_1 \leq \ldots \leq t_n$ and compute the covariance between $$\sum_{i=1}^n a_i \tilde B(t_i) \text{ and } B(1)$$
They see that the covariance is $0$ and hence they conclude that they're independent. Is it because $(\tilde B(t_1), \ldots , \tilde B(t_n))$ and $B(1)$ are jointly Gaussian?