Different interpretations of indicator random variable

Question

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.

Answer 1

Let me suggest to squarely forget these so-called definitions and to stick to the following, more canonical, version:

In a reference set $\Omega$, for every $A\subseteq\Omega$, the function $\mathbf 1_A:\Omega\to\mathbb R$ is defined by $\mathbf 1_A(\omega)=1$ if $\omega\in A$ and $\mathbf 1_A(\omega)=0$ if $\omega\in\Omega\setminus A$.

When the space $\Omega$ is measurable, that is, when one is given a sigma-algebra $\mathcal F$ on $\Omega$, then $\mathbf 1_A$ is measurable if and only if $A$ is in $\mathcal F$, as is easily seen since $A=(\mathbf 1_A)^{-1}(\{1\})$.

The function $\mathbf 1_A$ is often called the indicator function of the set $A$ by probabilists, its characteristic function by analysts, and it is also denoted by $\chi_A$.