## Bayesian Updating

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

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

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

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

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