ThmDex – An index of mathematical definitions, results, and conjectures.
Conditional probability of the sample space
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$
Then \begin{equation} \mathbb{P}(\Omega \mid \mathcal{G}) \overset{a.s.}{=} 1 \end{equation}
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$
This result is a particular case of R4338: Conditional probability of almost surely true event. $\square$