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

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) $$\mathbb{E} |X|, \mathbb{E} |Y| < \infty$$ (iii) $$X \overset{a.s.}{\leq} Y$$
Then $$\mathbb{E}(X) \leq \mathbb{E}(Y)$$

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) $$\mathbb{P}(X \leq Y) = 1$$
Then $$\mathbb{E}(X) \leq \mathbb{E}(Y)$$
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) $$\mathbb{E} |X|, \mathbb{E} |Y| < \infty$$ (iii) $$X \overset{a.s.}{\leq} Y$$
This result is a particular case of R1514: Isotonicity of signed basic integral. $\square$