In a Probability book by Karr, the Indicator random variable is defined as follows:
Indicator random variable. The indicator function of an event A is a random variable: for each B,
{$1_A\le t$} = $\begin{cases} \emptyset & \mbox{if 0 $\notin$ B, 1$\notin$ B} \\ A^c & \mbox{if 0 $\in$ B, 1$\notin$ B} \\ A & \mbox{if 0 $\notin$ B, 1$\in$ B} \\ \Omega & \mbox{if 0 $\in$ B, 1$\in$ B,} \end{cases}$
and in each case {$1_A\le t$} $\in \mathcal{F}$. Conversely, if A is subset of $\Omega$ such that $1_A$ is a random variable, then since A = {$1_A = 1$}, A is an event. Thus, indicator functions link events and random variables.
In my lecture notes, this is defined differently as follows:
Random indicators. For an event $A\in\mathcal{F}$, {$1_A\le t$} = $\begin{cases} \emptyset, & \mbox{$t<0$,} \\ A^c, & \mbox{$0\le t<1$,} \\ \Omega, & \mbox{$t\ge 1$.} \end{cases}$
I am not sure about the differences and the intuition behind them. For the textbook one, why in there is no $t$ in the description of the function, only B appears.
Any detailed explanations for each case (textbook and lecture notes) are greatly appreciated.