Why does the event $\{X \leqslant t\}$ is equal to the sample space when $t \geqslant c$ and $X=c$? This is what is written in my notes.  

Constants are RVs: for $X=c=const$, one has $\{X \leqslant t\} = \begin{cases} \emptyset, t<c \\  \Omega, t \geqslant c\end{cases}$

Why does the event $\{X \leqslant t\}$ is equal to the sample space when $t \geqslant c$ and $X=c$?
 A: It's simple, when $X=c$ and $t\geq c$:
$\{X\leq t\}=\{\omega\in \Omega| X(\omega)\leq t\}=^1=\{\omega\in \Omega| X(\omega)=c\leq t\}=^2=\Omega$
Explanations:
 1. $X(\omega)=c$ for any $\omega$ (this is what is meant by $X=c$).
2. Since $t\geq c$, then $X(\omega)=c\leq t$ for any $\omega$  
A: Let $\newcommand{\F}{\mathscr{F}}(\Omega,\F)$ be a measurable space, which means that $\Omega$ is a set and $\F$ is a sigma-field of subsets of $\Omega$. A random variable $X$ is a measurable function $X:\Omega\to\mathbb{R}$, meaning that it must be the case that for every Borel set $B$ in $\mathscr{B}$, where $\mathscr{B}$ is the usual Borel sigma-field of $\mathbb{R}$, the inverse image $X^{-1}(B)=\{\omega\in\Omega:X(\omega)\in B\}$ belongs to $\F$. It is possible to prove that to check that some $X$ is a random variable we don't have to check the inverse image of every Borel set. It is enough to check that sets of the form $\{\omega\in\Omega:X(\omega)\leq t\}$ belong to $\F$, for every $t\in\mathbb{R}$.
Your notes are probably stating that constant functions are real variables. Let $X:\Omega\to\mathbb{R}$ be defined by $X(\omega)=c_0$. We have two cases to check: if $t<c_0$, then $\{\omega\in\Omega:X(\omega)\leq t\}=\emptyset$, which belongs to $\F$, by definition. For $t\geq c_0$, $\{\omega\in\Omega:X(\omega)\leq t\}=\Omega$, which also, by definition, belongs to $\F$ (remember that $\emptyset$ and $\Omega$ must belong to any sigma-field of subsets of $\Omega$). This concludes the proof that constant functions are random variables.
Notice that we have not introduced a probability measure over $(\Omega,\F)$. Measurability is a property that does not make reference to specific measures over the considered measurable space.
