On Definition of Random Variables Given a probability space $(\Omega,  \mathcal{F}, P)$, a random variable X is defined to be a function from $\Omega$ to $ \mathbb{R}$ such that for every $a \in \mathbb{R}$, the set
$\{X \leq a\} := \{ω \in \Omega : X(\omega) ≤ a\}$ belongs to $\mathcal{F}$.
From the axiomatic definition of a probability space, for every $a \in \mathbb{R}$, if the set $\{X \leq a\}$ belongs to $\mathcal{F}$, the set $\{X > a\}$ must also belong to $\mathcal{F}$. Consequently, the set $\{a < X \leq b\}$ must also belong to $\mathcal{F}$.
I want to show that the set $\{a \leq X < b\}$ also belongs to  $\mathcal{F}$ as well, so I can generalize it to any other set, but I have not managed to prove it.
Anyone has an idea on how to do it?

To quote from Wikipedia, a $\sigma$-algebra is defined as

*

*$\Omega$ is in $\mathcal F$, and $\Omega$ is considered to be the universal set in the following context.

*$\mathcal F$ is closed under complementation: If $A$ is in $\mathcal F$, then so is its complement, $A^c$.

*$\mathcal F$ is ''closed under countable unions'': If $A_1$, $A_2,\ldots$  are in $\mathcal F$, then so is $A=A_1 ∪ A_2 ∪ \cdots$
From these properties, it follows that the σ-algebra is also closed under countable intersection (by applying De Morgan's laws).
