I know that "|" and ":" are both used to create sets and are read as "such that".
I now encountered ";". Since I didn't know its meaning, so I looked it up on wikipedia, where it says that it can be used interchangeably with the other two.
The problem is, this doesn't make much sense when I apply it to the example where I read it, namely the definition of a bayesian nash equilibria:
$$\sum_{t_{-i}\in T_{-i}}p_i(t_{-i}|t_i)u_i(s_i^*(t_i),s_{-i}^*(t_{-i});t_i)\geq \sum_{t_{-i}\in T_{-i}}p_i(t_{-i}|t_i)u_i(a_i,s_{-i}^*(t_{-i});t_i)$$
It just doesn't make sense to change notation style within one definition, right?
Could you tell me how I should read the ";" here?