ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4662 on D1748: Signed basic integral
Subresult of R3863: Signed basic integral with respect to a point measure

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. $\square$