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

Result R4754 on D3161: Random real number

Formulation 0

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$

Let $B \subseteq \mathbb{R}$ be a D5113: Standard real Borel set.

Then \begin{equation} \mathbb{P}(X \in - B) = \mathbb{P}(- X \in B) \end{equation}

Proofs

Proof 0

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$

Let $B \subseteq \mathbb{R}$ be a D5113: Standard real Borel set.

Result R4756: shows that $\{ X \in - B \} = \{ - X \in B \}$. The claim follows. $\square$