ThmDex – An index of mathematical definitions, results, and conjectures.
Isotonicity of real expectation
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $X, Y : \Omega \to \mathbb{R}$ are each a D3161: Random real number on $P$
(ii) \begin{equation} \mathbb{E} |X|, \mathbb{E} |Y| < \infty \end{equation}
(iii) \begin{equation} X \overset{a.s.}{\leq} Y \end{equation}
Then \begin{equation} \mathbb{E}(X) \leq \mathbb{E}(Y) \end{equation}
Formulation 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $X, Y : \Omega \to \mathbb{R}$ are each an D3066: Absolutely integrable random number on $P$
(ii) \begin{equation} \mathbb{P}(X \leq Y) = 1 \end{equation}
Then \begin{equation} \mathbb{E}(X) \leq \mathbb{E}(Y) \end{equation}
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $X, Y : \Omega \to \mathbb{R}$ are each a D3161: Random real number on $P$
(ii) \begin{equation} \mathbb{E} |X|, \mathbb{E} |Y| < \infty \end{equation}
(iii) \begin{equation} X \overset{a.s.}{\leq} Y \end{equation}
This result is a particular case of R1514: Isotonicity of signed basic integral. $\square$