Prove that an interval is NOT in the sigma algebra generated by a collection.

Question

I have been thinking about this for a while and could not find an answer. This is a homework question so any hints are appreciated (HINTS ONLY).
Let $\mathcal{M}$ be the sigma-algebra in $\mathbb{R}$ generated by the collection $$\mathcal{F} = \{ [n, n+1), n \in \mathbb{Z} \}.$$ Prove that the semi-open interval $[0, \frac{1}{2}) \notin \mathcal{M}$.

My approach was to find a sigma algebra $\mathcal{C}$ containing the collection $\mathcal{F}$ such that $[0, 1/2) \notin \mathcal{C}$.

Answer 1

Your approach is correct. The obvious choice of $\mathcal{C}$ would of course be $\mathcal{M}$.

Hint: $\mathcal{M}$ must contain all unions of the form $\bigcup_{n=1}^\infty A_n$, where $A_1,A_2,\dots \in \mathcal{F}$. Is $\{\bigcup_{n=1}^\infty A_n \: | \: A_1,A_2,\dots \in \mathcal{F}\}$ a $\sigma$-algebra?