# Understanding Bayes Rule of conditional expectation

Let $$\mathcal{F}$$ be a $$\sigma$$-algebra, $$P$$ and $$Q$$ be equivalent martingale measures and $$\frac{dQ}{dP}$$ the Radon Nikodym Derivative.
I learned that $$\Bbb{E}_Q[X]=\Bbb{E}_P[\frac{dQ}{dP}X]$$, which makes sense if one looks at the following: $$\Bbb{E}_Q[X]=\int_\Omega X dQ=\int_\Omega X \frac{dQ}{dP} dP=\Bbb{E}_P[\frac{dQ}{dP}X]$$Recently I was introduced to the Bayes Formula for conditional expectation, which states that $$\Bbb{E}_Q[X|\mathcal{F}]\;\; \Bbb{E}_P[\frac{dQ}{dP}|\mathcal{F}]=\Bbb{E}_P[\frac{dQ}{dP}X|\mathcal{F}]$$Comparing this with the version that I learned first, the term $$\Bbb{E}_P[\frac{dQ}{dP}|\mathcal{F}]$$ has been included and the only explanation I have is that $$\Bbb{E}_P[\frac{dQ}{dP}|\mathcal{F}]=\Bbb{E}_Q[1|\mathcal{F}]=1$$Is this understanding correct? And if so, why bother including the term on some occasions and excluding it on other occasions? Thank you for clearing up my confusion!

• Have a look of here. – Gordon Jan 13 at 18:09