Hello,

I have been given the follwoing subsets of $\displaystyle \mathbb{R}$: $\displaystyle A=\left[0,1 \right]$, $\displaystyle B=\left]\tfrac{1}{2},\infty \right[$ and $\displaystyle C= \mathbb{Z}$.

I wish to show that the sets $\displaystyle \left\{-1,-2,-3,... \right\}$, $\displaystyle \left\{0\right\}$ and $\displaystyle A=\left[0,\tfrac{1}{2}\right] \cup \left\{1\right\} $ belong to the $\displaystyle \sigma$-algebra $\displaystyle \sigma(\left\{A,B,C \right\})$.

Would it be enough to, tediously, write out the $\displaystyle \sigma$-algebra and check that the subsets are indeed included or is there an easier method?

Thanks.