In Zdzislaw Brzezniak and Tomasz Zastawniak, "Basic Stochastic Processes"
A random variable is defined as,
How do I check the condition
For every Borel set $B \in \mathcal{B}(\mathbb{R})$
given that this set is very large?
Note: The textbook defines a Borel set as the,
As a concrete example, let the space $\Omega = \{H,T\}$ denoting Head and Tail respectively. Then the $\sigma$-field on $\Omega$ is the set $\mathcal{F} = \{\varnothing, H, T, \{H,T\}\}$
Let's define a random variable which goes from $\Omega$ to $\mathbb{R}$.
$\xi(H) = 1, \xi(T) = -1$
How do I check that $\{\xi \in B\} \in \mathcal{F}$ for every single set in the Borel set?
You don't have to. There is a measurability criterion (I believe it should be also in the textbook you read): it is enough to check that this is satisfied for any set from a family generating $\mathcal{B} (\mathbb R)$ e.g. for $\{(-\infty, x), x\in \mathbb R\} $. Also you don't need to employ the definition each time you need to show the measurability : there are various facts concerning transformations of random variables.