Lets suppose we have $\Omega = [0,1]$ and let us have $\sigma$-algebra $F=\{\emptyset, [0,\frac{1}{3}],(\frac{1}{3},1],[0,1]\}$ and the function X is defined as $$ X(w)= \begin{cases} 3, \text{if }w\in[0,\frac{1}{3}]\\ -5, \text{if }w\in(\frac{1}{3},1] \end{cases} $$
Now my question is if I add more values to the random variable will it still be measurable w.r.t. to $F$ above?