In Tenenbaum's Introduction to Analytic and Probabilistic Number Theory, he defines Expectation of an arithmetic function $f(n)$ with respect to uniform distribution $v_N$ by
$\mathbb{E_N}(f) := \int_{-\infty}^{\infty} z \;d\mathbb{F_N}(z) = \frac{1}{N}\sum_{n\leq N} f(n)$
Here, $\mathbb{F_N}(z) = \frac{1}{N} \# \{n \leq N: f(n) \leq z\}$
How do you get the expression with summation from the expression with integral in the definition?