What is the measure of $[x, x+\frac{1}{n})$?

Question

Suppose $x \in \mathbb{R}$ and let $n \in \mathbb{N}$

How could we interpret $\lim\limits_{n \to \infty}$ measure $\{[x,x+\frac{1}{n})\}$ ?

Initially, we might be tempted to think that $x+\varepsilon \to x$ as $n \to \infty$ in which case our set becomes infinitely close to $\{x\}$.

Then, another idea is to consider the infemum of $[x,x+\varepsilon)$ which is just $x$ but that takes us back to the above statement.

Are there any ideas on how to measure this set as $n$ increases arbitrarily?

I would say the measure of $[x,x+\frac1n)$ is $\frac1n$, which approaches $0$ as $n\to\infty$ — J. W. Tanner, Apr 22 '21 at 01:45

score 1 · Accepted Answer · answered Apr 22 '21 at 01:46

1

Continuity from above: Let $E_n$ be a decreasing sequence of measurable sets with some set with finite measure. Then $m (E_n)$ converges to $m(E)$ where $E$ is the limit of the sequence.

Note that the sequence you have is a decreasing one, and every set has a finite measure. I think it helps. Please take a look.

Continuity from below and above

answered Apr 22 '21 at 01:46

Matha Mota

3,472

Whether or not they have finite measure really depends on the measure they are using – Arctic Char Apr 22 '21 at 01:53
Yes. That's right. I considered Lebesgue measure here. Thanks for pointing out the issue. – Matha Mota Apr 22 '21 at 01:55

What is the measure of $[x, x+\frac{1}{n})$?

1 Answers1