[Math] How to determine if a random variable is $\mathcal F$-measurable

Question

For example :

Consider the state space $\Omega = \mathbb{R}$, the $\sigma$-algebra, $\mathcal{F} = \{(-\infty, 0], (0, \infty), 0, \mathbb{R}\}$ and the random variable $X : \Omega \rightarrow \mathbb{R}$ defined by
\begin{align*}
X(\omega) = \begin{cases}
3 & \omega < 0\\
5 & \omega \geq 0
\end{cases}
\end{align*}
Is $X$ $\mathcal{F}$-measurable?

Answer 1

Just check if $\{\omega:X(\omega)<c\}\in \mathcal{F}$, $c \in \mathbb{R}$.

1. $c<3$. In this case, since no $\omega$ satisfies, $\{\omega:X(\omega)<c\}=\varnothing\in \mathcal{F}$

2. $3 \leqslant c<5$. In this case, $\{\omega:X(\omega)=3\}=(-\infty,0) \in \mathcal{F}$

3. $c \geqslant 5$. In this case, $\{\omega:X(\omega)=5\}=[0,+\infty)=0\cup(0,+\infty)\in \mathcal{F}$

So $X$ is $\mathcal{F}$-measurable.