Prove conditional expectation $E[X\mid X>x]$ is the unconditional expectation $E_{P^*}[X]$ under a probability measure $P^*$

Question

Prove that the conditional expectation $\mathbb E[X\mid X>x]$ (here x is fixed, say x=10) is the unconditional expectation $\mathbb E_{\mathbb P^*}[X]$ under a probability measure $\mathbb P^*$. Derive a formula for $\mathbb P^*$ in terms of the original probability measure $\mathbb P$.

I can only think of taking double expectation on the conditional expectation to remove the condition, but I'm not sure if that's the right step to take. Any suggestion is appreciated.

Answer 1

$\mathbb{P^*}$ is defined as follows: $$\mathbb{P^*}(A) := \mathbb{P}(A\mid X > x) = \frac{\mathbb{P}(A\cap \{X > x\})}{\mathbb{P}(X > x)}, ~~~\forall A \in \mathcal{F}.$$

Of course, this definition is only valid when $\mathbb{P}(X > x) > 0$. It can be checked $\mathbb{P^*}$ defined in this way is indeed a probability measure on $(\Omega, \mathcal{F})$ and satisfies $\mathbb{E}(X\mid X > x) = \mathbb{E}_{\mathbb{P^*}}(X)$.