Almost Sure Inequalities Between Bounded Random Variables Pass to Expectations

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $K\ge0$ be a real number, and let $X$ and $Y$ be random variables on it with $|X(\omega)|\le K$ and $|Y(\omega)|\le K$ for every $\omega\in\Omega$. Suppose there is an event $\Omega_*\in\mathcal{F}$ with $P(\Omega_*)=1$.

1. (Domination.) If $X(\omega)\le Y(\omega)$ for every $\omega\in\Omega_*$, then $\mathbb{E}[X]\le\mathbb{E}[Y]$.

2. (Equality.) If $X(\omega)=Y(\omega)$ for every $\omega\in\Omega_*$, then $\mathbb{E}[X]=\mathbb{E}[Y]$.

3. (Bound.) If $c\ge0$ is a real number and $|X(\omega)|\le c$ for every $\omega\in\Omega_*$, then $|\mathbb{E}[X]|\le c$.

4. (One-sided version.) Let $K'\ge0$ be a real number, let $W$ be a random variable with $W(\omega)\ge-K'$ for every $\omega\in\Omega$, and let $U$ be a random variable with $|U(\omega)|\le K$ for every $\omega\in\Omega$ such that $|W(\omega)|\le U(\omega)$ for every $\omega\in\Omega_*$. Then $W$ is integrable and $|\mathbb{E}[W]|\le\mathbb{E}[U]$.

Here $\mathbb{E}$ denotes the expectation, which for a bounded random variable exists because $P$ is a finite measure.