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Result R4823 on D2012: Conditional probability
Subresult of R4339: Conditional probability of almost surely false event

Conditional probability of the empty event

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

$\begin{equation} \mathbb{P}(\emptyset \mid \mathcal{G}) \overset{a.s.}{=} 0 \end{equation}$

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 $\Omega$

$\square$