Suppose I have this list $1, 1, 1, 1, 1…$ with $\bf{Ord}$ (the proper class of ordinals) number of $1$'s. Then would the average (arithmetic mean) of the list be $1$ because there is literally no other values in the list?
A good way is to do
$$\lim_{n \rightarrow \alpha} \frac {[\Sigma a_i\rightarrow\alpha]} {n}$$
where $\alpha$ is the ordinal length of the summed sequence (let it grow! let it grow! let it grow! ), $+\infty$ if it is infinite, and $[\Sigma a_i\rightarrow\alpha]$ is the sum of the first $\alpha$ terms of the sequence; and take the supremum across all ordinals. However, the order of the sequence may affect the result. Or I can just sort the list, make a bijection between $\bf{Ord}$ and the list, then reorder the list with the bijection (tell me if there are any 'problems' with this).
Or I can integrate, but a 'good' definition for integration on proper-clasd-sized lists, for example the surreal numbers is still being worked on or not talked about at all (as far as I know).
So, are there any popular ways of calculating the arithmetic mean of a proper-class-sized list of real numbers? If so, what are some of them? If not, is this even well-defined?