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How do I prove a half open interval is a sigma algebra?

Mathematics Asked by Marcelo RM on November 6, 2021

I was reading the lecture notes about probability and random processes, Lecture Notes, then I’m stucked here:

Example 1.3.6 Let $Omega = mathbb R$ and suppose that we have a sigma field A such that all intervals of the form:

$[ 1, 2 – 1/n) in mathcal{A}$

So, I would like to know how to prove that interval is a $sigma$ algebra?

  1. $Omega in mathcal{A}$
  2. If $A in mathcal{A}$, then $A^{c} in mathcal{A}$
  3. If $A_{n} in mathcal{A}$ for each $n$ in a countable collection $(A_{n})_{n=1}^{infty}$, then $cup_{n=1}^{infty} in mathcal{A}$.

I know from the lecture note that:

$[1, 2) in mathcal{A}$, since $lim_{ntoinfty} [1, 2 – 1/n)$

I think its prove statement 3. Am I right?

But I don’t know how to prove statements 1 and 2.

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