I am trying to spot the mistake in the following simple equation:
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a prob. space, $(\mathcal{F}_n)_n \nearrow \mathcal{F}$ and let $\mathbb{Q} \approx \mathbb{P}$, thus $\mathbb{Q}$-a.s. is equivalent to $\mathbb{P}$-a.s.. For a $X \in L^\infty$ (thus we can freely apply the dominated convergence theorem) we have
\begin{align*} \mathbb{E}_\mathbb{P}[X] &= \mathbb{E}_\mathbb{P}[\lim_n\mathbb{E}_\mathbb{Q}[X | \mathcal{F}_n]] = \lim_n\mathbb{E}_\mathbb{P} [ \mathbb{E}_\mathbb{Q}[X | \mathcal{F}_n]]\\ &= \lim_n \mathbb{E}_\mathbb{P} [\mathbb{E}_\mathbb{P}[d\mathbb{Q}/ d\mathbb{P} \text{ } \text{ } X | \mathcal{F}_n]] = \lim_n \mathbb{E}_\mathbb{P} [ d\mathbb{Q}/ d\mathbb{P} \text{ } \text{ } X] = \mathbb{E}_\mathbb{Q}[X] \end{align*}
I really cannot see the mistake, and I also think that this cannot be true. I would be really grateful if you could help me with this.