Could some please explain what does notation $s \in \{ t\in [q,1): 0
$Var(g(s)^{Z_{1}}>0$ for any $s\in\{t\in [q,1):0<g(t)<1\}\supset(q,1)$
where $g(s)$ is in an inverse of probability generating function and $Z_1$ is a random variable.
I am not sure if I understand $s \in \{ t\in [q,1): 0<g(t)<1\}\supset (q,1)$ in a right way. So $s$ is some point from the set of points for which $0<g(t)<1$ true is.  However what does $\{ t\in [q,1): 0<g(t)<1\}\supset (q,1)$ exactly mean, do we mean, that the interval is a subset of the set of points?
 A: Let's take it bit by bit.

*

*$t\in[q,1)$ means $t$ is in the half-open interval, i.e. it satisfies $q\leq t<1$.


*$\{t\in[q,1):0<g(t)<1\}$ is the set of points in that half interval where the function $g$ is strictly between $0$ and $1$.


*$\{t\in[q,1):0<g(t)<1\}\supset(q,1)$ means that this set contains all $t$ with $q<t<1$. Consequently, the only number remaining we don't know about is $q$ itself - the set is either $[q,1)$ or $(q,1)$. This assumes the writer is using the convention that $\supset$ means "contains or is equal to".*


*$s\in\{t\in[q,1):0<g(t)<1\}$ simply means $s$ satisfies both properties, i.e. $q\leq s< 1$ and $0<g(s)<1$.
The whole thing is just asserting both of the last two points in a more compact (but harder to read) form.

* This is common but IMO misguided, since $\supseteq$ also means "contains or is equal to", and using $\supset$ for this means that if you ever want to talk about strict containment you have to resort to using $\supsetneqq$.
A: It seems given your comments that this means (I suppose that $q$ is less than $1$):

$s$ belongs to the set of real numbers between $q$ and $1$ ($1$
excluded) let's call them $t$, such that $g(t)$ is part of the open
set $(0,1)$. This set contains the open set $(q,1)$.

This is generally a dirty way to say that any real number $x\in(q,1)$ satisfies
$$0<g(x)<1$$ Therefore, by picking $s$ in the open interval $(q,1)$, we are guaranteed to have both inequalites, which in the context of probability theory may be useful.
Again with little to no context, it is hard to get what the author means.
