intervals as unions and intersections

Sep 2016
15
0
Denmark
Hello,

I have the following set:

\(\displaystyle \mathcal{J}= \left\{ (c,c+1), c \in \mathbb{R} \right\}\)

I would like to show that the sets:

\(\displaystyle (0, \frac{1}{2})\) , \(\displaystyle (0, \frac{1}{2}]\) , \(\displaystyle \left\{\frac{1}{2} \right\}\) , \(\displaystyle [0,1]\)

belong to the \(\displaystyle \sigma\)-algebra \(\displaystyle \sigma(\mathcal{J})\).

The elements of \(\displaystyle \sigma(\mathcal{J})\) are obtained by countable unions/intersections/differences of the elements of \(\displaystyle \mathcal{J}\).

I am having trouble constructing these intersections and unions. Assistance would be appreciated.
 
Jan 2009
422
117
Not familiar with σ-algebra, but playing around with the following elements of J seemed useful: (-1,0), (0, 1), (-1/2, 1/2), (1/2, 3/2), (1, 2)

ex. (0, 1/2) = (-1/2, 1/2) intersect (0, 1)

I'll let you try the others on your own