I am having a hard time with the intuition behind some of the representation theorems dealing with Brownian Motion. I think if someone can simply explain the intuition behind this theorem then everything will fall into place:
Theoreom:
For any $0\le a <b$ and any finite random variable $X\in\mathscr{F}_a$, there is a stopping time $\tau$ with $a\le\tau<b$ such that \begin{align*}X=\int_a^{\tau}\frac{1}{b-t}dB_t\end{align*}
That is the theorem. One thing I am having a massive problem with is understanding how this can be true since the right side of the above equality is a Gaussian process, but I don't see why $X$ needs to be a Gaussian Process. Can anyone explain this?