# Samples in Measure Theory

What does the sentence "Let \(x_1, \dots, x_n\) be \(n\) samples from the probability distribution \(f\)" mean mathematically?

Let's define the probability density \(f: \mathbb{R} \to \mathbb{R}\) such that \(\int_\mathbb{R} f(\mu) \mathrm{d}\mu = 1\). \(f\) is a density corresponding to a measure \(\mathbb{P}\) with respect to, e.g., the Lebesgue measure on \(\mathbb{R}\). The rigorous meaning of the above statement is that \(x_1, \dots, x_n\) are independent *random variables* with the same associated measure and distribution \(f\). For any \(r_1, \dots, r_n \in \mathbb{R}\), the probability that \(x_i < r_i\) is true for all \(i=1, \dots, n\) is

\[ \prod_{i=1}^n \int_{-\inf}^{r_i} f(x) \mathrm{d}x \]