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

Result R4512 on D1748: Signed basic integral

Formulation 0

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: 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}

Let $[a, b] \subseteq [-\infty, \infty]$ be a D4658: Closed basic interval such that

(i)	\begin{equation} a \overset{a.s.}{\leq} X \overset{a.s.}{\leq} b \end{equation}

Then

(1)	\begin{equation} a \leq \mathbb{E} X \leq b \end{equation}
(2)	\begin{equation} \mathbb{E} X = a \quad \iff \quad X \overset{a.s.}{=} a \end{equation}
(3)	\begin{equation} \mathbb{E} X = b \quad \iff \quad X \overset{a.s.}{=} b \end{equation}

Formulation 1

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: 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}

Let $[a, b] \subseteq [-\infty, \infty]$ be a D4658: Closed basic interval such that

(i)	\begin{equation} \mathbb{P}(X \in [a, b]) = 1 \end{equation}

Then

(1)	\begin{equation} \mathbb{E} X \in [a, b] \end{equation}
(2)	\begin{equation} \mathbb{E} X = a \quad \iff \quad X \overset{a.s.}{=} a \end{equation}
(3)	\begin{equation} \mathbb{E} X = b \quad \iff \quad X \overset{a.s.}{=} b \end{equation}

Proofs

Proof 0

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: 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}

Let $[a, b] \subseteq [-\infty, \infty]$ be a D4658: Closed basic interval such that

(i)	\begin{equation} a \overset{a.s.}{\leq} X \overset{a.s.}{\leq} b \end{equation}

This result is a particular case of R4500: . $\square$