ThmDex – An index of mathematical definitions, results, and conjectures.
Result R2090 on D198: Probability measure
Isotonicity of probability measure
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space.
Let $E, F \in \mathcal{F}$ each be an D1716: Event in $P$ such that
(i) $E \subseteq F$
Then \begin{equation} \mathbb{P}(E) \leq \mathbb{P}(F) \end{equation}
Formulation 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space.
Then \begin{equation} \forall \, E, F \in \mathcal{F} \left( E \subseteq F \quad \implies \quad \mathbb{P}(E) \leq \mathbb{P}(F) \right) \end{equation}
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space.
Let $E, F \in \mathcal{F}$ each be an D1716: Event in $P$ such that
(i) $E \subseteq F$
This result is a particular case of R975: Isotonicity of unsigned basic measure. $\square$