How to denote an event that is always true?

Question

What is a good symbol to denote an event that is always true? It is a good way to use the symbol of the probability space, e.g. $\Omega$. But in my paper, the probability space is very complicated and I do not plan to discuss explicitly about it. For a normal event, I denote it with its condition, e.g. $\{X>0\}$. But for an event that is always true in the probability space, I cannot find a good way to denote it — it also looks confusing to use something like $\{1>0\}$.

Answer 1

Here are two admittedly somewhat contrived variants:

• The empty condition $$\{\}$$ is vacuously true.

• The Iverson brackets are defined as \begin{align*} [[P(x)]]=\begin{cases} 1&\qquad P(x) \ \text{ true}\\ 0&\qquad P(x) \ \text{ false} \end{cases} \end{align*}

  A statement which is always true can be written as $[[1]]$. So, instead of $\{1>0\}$ we could also write \begin{align*} \{[[1]]\} \end{align*}