ThmDex – An index of mathematical definitions, results, and conjectures.
Signed basic expectation with respect to a point probability measure
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P}_{\omega_0})$ be a D5672: Point probability space such that
(i) $X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$
(ii) \begin{equation} \mathbb{E} |X| < \infty \end{equation}
Then \begin{equation} \mathbb{E}_{\mathbb{P}_{\omega_0}} X = X(\omega_0) \end{equation}
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P}_{\omega_0})$ be a D5672: Point probability space such that
(i) $X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$
(ii) \begin{equation} \mathbb{E} |X| < \infty \end{equation}
This result is a particular case of R3863: Signed basic integral with respect to a point measure. $\square$