Let $Z \in \text{Gaussian}(0, 1)$ be a D211: Standard gaussian random real number.
Let $B \in \mathcal{B}(\mathbb{R})$ be a D5113: Standard real Borel set.

Then \begin{equation} \mathbb{P}(Z \in B) = \int_B \frac{1}{\sqrt{2 \pi}} e^{- t^2 / 2} \, d t \end{equation}

Let $Z \in \text{Gaussian}(0, 1)$ be a D211: Standard gaussian random real number such that

(i)	$\mu_Z$ is a D204: Probability distribution measure for $Z$

Let $\ell$ be the D5645: Real Lebesgue measure.
Let $t \in \mathbb{R}$ be a D993: Real number.

Then \begin{equation} \frac{d \mu_Z}{d \ell} (t) = \frac{1}{\sqrt{2 \pi}} e^{- t^2 / 2} \end{equation}