Notation for length of arbitrarily long sequences in a set Suppose I have the set of arbitrarily long sequences of dice rolls:
\begin{equation}
\Omega = \bigcup_{n = 1}^\infty [6]^n = \{ (1),(2),(3),(4),(5),(6),(1,1),(1,2),\ldots \}
\end{equation}
Given some sequence $\omega \in \Omega$, what's the notation to get the length of the sequence? Is it $|\omega|$?

I have seen each of the following notations used to represent the length of a finite sequence $\sigma$:


*

*$\vert\sigma\vert$

*$length(\sigma)$

*$lh(\sigma)$
Personally, I think all three are perfectly fine, although it's worth spending a sentence saying what your notation means.
Note that "$\vert\sigma\vert$" is actually perfectly correct for finite sequences: a sequence is a set of ordered pairs, so it's length is indeed its cardinality. That said, once we look at infinite sequences this breaks down - a sequence of ordertype $\mathbb{N}$ and a sequence of ordertype $\mathbb{N}+1$ have the same cardinality. Since I often deal with infinite sequences, I tend to prefer $length$ or $lh$.
