ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4756 on D529: Map inverse image

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 $$\{ X \in - B \} = \{ - X \in B \}$$

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 $$\{ \omega \in \Omega : X(\omega) \in - B \} = \{ \omega \in \Omega : - X(\omega) \in B \}$$
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.
This result is a particular case of R4755: Inverse image of a reflected real set. $\square$