I am working through a problem where I wish to claim that: $$ \mathbb E \left[\prod_{i=0}^n \frac{f(X_i)}{\mathbb E [f(X_i) \ | \ \mathcal F_i]} \right] = \mathbb E \left[\prod_{i=0}^n \frac{f(X_i)}{f(X_i)} \right] = 1. $$
Is this a valid application? The $X_i$ random variables are not necessarily independent. (of course it would be trivial then)
If $X_i$ is independent of $\mathcal F_i$ for each $i$ then the left side becomes $\frac {\mathbb E \prod_i {f(x_i)}} {\prod_i \mathbb E f(X_i)}$ which need not be $1$ since $X_i$ are not assumed to be independent.
On the other hand if $(X_i)$ is adapted to $(\mathcal F_i)$ then the equation is obviously true.