# Notation: Set where function is positive

Given some function $f: X \to \mathbb{R}$ from some space $X$ to the reals.

I have sometimes seen the notation $\{f > 0\}$ used to denote the set $\{x \in X : f(x) > 0 \}$. Is this a common/usual notation?

I would like to use the shorter one because it is easier to write. But the longer one is more clear. Which one is more natural?

• You answered yourself. The longer one is more clear and the shorter one is easier to write. And yes the shorter one is common ! – alkabary Jul 10 '15 at 23:23
• @alkabary ok perfect, thank you :) – Loreno Heer Jul 10 '15 at 23:23
• You could also write $f^{-1}((0,\infty))$. – Ben Whitney Jul 10 '15 at 23:40
• Notice that the shorter notation is very common in fields like measure theory, probabilistic theory, and related fields. It less common in general. – user251257 Jul 11 '15 at 0:19
• @BenWhitney: Why not simply $f^{-1}(\mathbb R^+)$? – celtschk Jul 11 '15 at 10:18

## 1 Answer

The shorter notation $\left\{f > 0\right\}$ is shorter and is very common in fields such as measure theory. It is less common but still occasionally seen in other fields. It's obviously not as clear as the longer notation.

You can safely use the shorter notation in most situations without causing any confusion, if you have any doubts about it, then go with the longer notation. Although that most likely won't be required in most scenarios.