For a certain $\sigma$-algebra $A$ on the real line, I would like to show that it contains the Borel sets. I can show that $A$ contains the left and right half-line $(a,\infty)$ and $(-\infty,b)$ for any real numbers $a$ and $b$. My question is : can I infer that $A$ contains the Borel sets by only prooving that it contains the left half-line or is it mandatory to show that $A$ contains both half-line? I'm not clear on how the Borel sets are generated from half-line and open intervals.
What is the complement of one of these half-lines? Then consider intersections, unions, etc. In other words, yes, if you can show your $\sigma$-algebra contains $(a, \infty)$ for any $a$, that is enough... but it sounds like it would be a good exercise for you to prove this. Here are some hints: