Let $X_1, X_2, X_3, \ldots \in \text{Random}(\mathbb{R})$ 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 $\mu \in \mathbb{R}$ be a D993: Real number.

Then the following statements are equivalent

(1)	\begin{equation} \lim_{N \to \infty} \sum_{n = 1}^N \frac{X_n - \mu}{N} \overset{a.s.}{=} 0 \end{equation}
(2)	\begin{equation} \mathbb{E} |X_1| < \infty \text{ and } \mu = \mathbb{E} X_1 \end{equation}

Let $X_1, X_2, X_3, \ldots \in \text{Random}(\mathbb{R})$ 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 $\mu \in \mathbb{R}$ be a D993: Real number.

Then the following statements are equivalent

(1)	\begin{equation} \mathbb{P} \left( \lim_{N \to \infty} \sum_{n = 1}^N \frac{X_n - \mu}{N} = 0 \right) = 1 \end{equation}
(2)	\begin{equation} \mathbb{E} |X_1| < \infty \text{ and } \mu = \mathbb{E} X_1 \end{equation}