Consider $ \Omega = (0,\infty)$ , $ \mathcal F = B(\Omega)$ , and $ \lambda > 0$ . Let $ P$ be the probability measure on $ (\Omega, \mathcal F)$ satisfying $ P((a,b]) = e^{-\lambda a} – e^{-\lambda b}$ . Also set $ X(\omega) = \omega$ and $ Y(\omega) = \min \{ \omega, \kappa \}$ for $ \kappa > 0$ . Compute

- $ E(X \mid \sigma (Y))$
- $ E(e^{-\alpha X} \mid \sigma (Y)), \quad \alpha >0 $

I have no idea how to start here. I realise that the function $ Y(\omega)$ is a piecewise linear function on $ (0,\kappa)$ and $ [\kappa, \infty)$ . Hence the $ \sigma$ -algebra $ \sigma (Y)$ is given by the collection of sets

$ \varnothing, \Omega$ , $ \{ B : B \subseteq (0,\kappa)\}$ , $ \{B : B \subseteq [\kappa, \infty)\}$ ,

but apart from that I have no clue… Any one that can provide me with a hint?