§ Splitting $f(x) = y$ into indicators

If the output of $f(x)$ is a natural number, then we can write the value $f(x)$ as:

$$f(x) = \sum_{i=1}^\infty [f(x) \geq i]$$

where $[f(x) \geq i]$ is $1$ if the condition is true and $0$ otherwise. Another useful indicator type equation is:

$$\sum_x f(x) = \sum_x \sum_i i \cdot [f(x) = i] = \sum_i i \cdot (\sum_x [f(x) = i])$$