Let $X, Y \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that

(i)	\begin{equation} \mathbb{P}(X \neq Y) = 0 \end{equation}

Then \begin{equation} X \overset{d}{=} Y \end{equation}

Let $X, Y \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that

(i)	\begin{equation} \mathbb{P} \{ \omega \in \Omega : X(\omega) \neq Y(\omega) \} = 0 \end{equation}

Let $B \in \mathcal{B}(\mathbb{R})$ be a D5113: Standard real Borel set

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