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I am asking about this question:

Checking an example of a monotone class.

1-Why in the example of the OP $(-\infty, a) \cup [a, \infty)$ not equal to $\mathbb{R}$?

2-How can I prove rigorously that $\{0\}$ does not belong to $$\{(-\infty, a_{n}), (-\infty, a_{n}], \emptyset, [a_{n}, \infty),\mathbb{R}, (a_{n}, \infty)\}, $$I believe that the answer of Kavi there is very convincing but is not their a rigorous proof?

Emptymind
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1 Answers1

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You have misunderstood that question. The claim made in the question is that $$\mathcal{G} = \{(-\infty, a), (-\infty, a], \emptyset, [a, \infty), (a, \infty): a \in \mathbb{R}\}$$ is itself a monotone class. The observation you make that $\mathbb{R} = (- \infty, a) \cup [a, \infty)$ is exactly the problem that the answer points to. If $\mathcal{G}$ were a monotone class then this would imply that $\mathbb{R} \in \mathcal{G}$. However $\mathbb{R} \not \in \mathcal{G}$ so that $\mathcal{G}$ cannot be a monotone class.

As for the second question you ask, from the definition of $\mathcal{G}$, if $A \in \mathcal{G}$ then there is $a \in \mathbb{R}$ such that $A \in \{(-\infty,a), (-\infty,a], (a, \infty), [a, \infty)\}$. So since $\{0\}$ is non-empty, you just need to check it is not one of these half-intervals for some $a$. This is obvious since all of the half-intervals contain uncountably many elements.

Rhys Steele
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  • so this example is also correct? and what about the increasing and decreasing condition of the monotone class for this example, is it satisfied? if so, how? could you please explain this point? – Emptymind Feb 23 '20 at 15:38
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    $\mathcal{G}$ is not a correct example as it is not a monotone class. $\mathcal{G}' = \mathcal{G} \cup \mathbb{R}$ is a monotone class that is closed under complements but is not a $\sigma$-algebra. Checking the increasing and decreasing conditions is just casework. E.g. an increasing family of sets $A_n$ in $\mathcal{G}$ is either of the form $A_n \in {(-\infty,a_n), (- \infty, a_n]}$ for every $n$ for some increasing sequence $a_n$ or of the form $A_n \in {(b_n, \infty), [b_n, \infty)}$ for every $n$ for some decreasing sequence $b_n$. It is straightforward to work through these cases – Rhys Steele Feb 23 '20 at 16:27
  • And what is the union of the increasing family is it inf a_{n} and sup b_{n}? – Emptymind Feb 25 '20 at 01:25
  • It depends on the exact sequences $a_n$ or $b_n$. Say we are in the case where $A_n \in {(-\infty, a_n), (-\infty,a_n]}$. Then you will have $\bigcup_n A_n \in { (- \infty, \sup a_n), (- \infty, \sup a_n]}$ depending on whether or not $a_n$ has a constant tail. The other case is similar but with the union being an interval with left endpoint $\inf b_n$. (notice that the $\inf$ and $\sup$ are the other way around to what you suggest) – Rhys Steele Feb 25 '20 at 12:19