Ok , Lets say there is some random variable $\displaystyle \omega$ that has a normal distribution N(0,1). Now suppose I receive a signal s such that
$\displaystyle s=\omega + \epsilon$ where $\displaystyle \epsilon$~$\displaystyle N(0,\sigma^2_{\epsilon})$.

Now, I know that $\displaystyle E[\omega | s] = W*s$ where $\displaystyle W=\frac{1}{1+\sigma^2_{\epsilon}}$

My question is, suppose I didnt know s, but I knew that s what in some set I. Does it follow that:

$\displaystyle E[\omega | s \in I] = \int_{I}W*sdF(s)$

My inital calculation and guess is that the answer is yes, but wanted to see what you all thought.