Set builder notation: Are variables that aren't specified after the "such that" colon considered to be parameters?

Question

Admission: I have no idea what I'm doing

I want to describe the following set (the set containing all probability distributions) in a more concise way

$$ D = \{ \{ ( t_1, P(t_1 | s)), ( t_2, P(t_2 | s)), ... \text{ for all } t_i \in T \} \text{ for all } s \in S \} $$

Is there a nice way to describe this set either by set builder notation or functions? Malice Vidrine suggests

$$ D = \{ \{ ( t, P(t | s)) : t \in T \} : s \in S \} $$

Are variables that aren't specified after the "such that" colon considered to be parameters?

Answer 1

There are two issues with the expression defining $D$.

The first one is the notation $t_i$, which suggests that you are working with a family, something like $(t_i)_{1 \leqslant i \leqslant n}$ or $(t_i)_{i \in I}$. If you are working with a set $T$, the indices $i$ are useless. This is why Malice Vidrine (whom you unfortunately quote incorrectly) suggested to write $\{\{(t, P(t|s)):t \in T\}:s \in S\}$.

The second issue is the double level in the set builder notation. Although this is mathematically correct, it would be more convenient to write

For each $s \in S$, let $D_s = \left\{\left(t, P(t|s)\right):t \in T\right\}$ --- or, if you are working with a family $(t_i)_{i \in I}$, $D_s = \left\{\left(t_i, P(t_i|s)\right):i \in I\right\}$. Then you can simply set $D = \{D_s : s \in S\}$.