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.