1
$\begingroup$

What is the intuition for the event $\{X>0\}$ being equivalent to $\cup_{n\in\mathbb{N}}\{X\geq\frac1n\}$? I have seen rigorous expositions on it, but have yet to develop a good inuition of this. Why just the natural numbers, i.e. why not $\cup_{n\in\mathbb{R}_{\geq0}}\{X\geq\frac1n\}$, and why don’t we use strict inequalities? When is it useful to write it this way? I have seen it in multiple proofs (e.g. for proving certain quantities are stopping times, proving non-negative r.v. with zero expectation is zero a.s.). but I haven't really seen why only including the natural numbers works. Cheers! (Also, please edit the tags if need be, not sure which ones to include!)

$\endgroup$
1
  • $\begingroup$ The equality is still true if you change the natural to the positive reals (not including 0), and it is still true also if you change the relaxed inequality by strict inequality. Have you tried to prove it showing the equality of the sets $\{x\in\mathbb{R}\mid x>0\}$ and $\cup_{n\mathbb N}\{x\in\mathbb R\mid x\geq \frac{1}{n}\}$? $\endgroup$
    – Javi
    Nov 29, 2020 at 14:09

1 Answer 1

6
$\begingroup$

The reason one uses natural numbers and not real numbers is to make sure we have a countable union. The countable union of measurable sets is measurable, but this need not be true for uncountable unions.

Also, you can also use strict inequalities, it doesn't change anything.

Ultimately, the 'intuition' here is the archimedian property: If $x >0$, you can choose $n$ so large that $x > 1/n$.

$\endgroup$
2
  • 1
    $\begingroup$ beautiful answer, thank you! $\endgroup$
    – user107224
    Nov 29, 2020 at 15:07
  • $\begingroup$ You are welcome! $\endgroup$
    – J. De Ro
    Nov 29, 2020 at 15:41

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .