Let $X_1, X_2, X_3, \ldots \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, X_2, X_3, \ldots$ is an D3358: I.I.D. random collection

Let $x \in \mathbb{R}$ be a D993: Real number.

Then \begin{equation} \lim_{N \to \infty} \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \leq x \}} \overset{a.s.}{=} \mathbb{P}(X_1 \leq x) \end{equation}

Let $X_1, X_2, X_3, \ldots \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, X_2, X_3, \ldots$ is an D3358: I.I.D. random collection
(ii)	$F : \mathbb{R} \to [0, 1]$ is a D205: Probability distribution function for $X_1$

Let $x \in \mathbb{R}$ be a D993: Real number.

Then \begin{equation} \mathbb{P} \left( \lim_{N \to \infty} \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \leq x \}} = F(x) \right) = 1 \end{equation}