I started reading "An introduction to measure theory" by Terence Tao. On page 23 on a pdf reader (pg 7 in the actual document), we are asked to think of an example of a set $E\subset$ $\require{enclose} \enclose{horizontalstrike}{\mathbb{R}}$ $\mathbb{R}^d$ which measure cannot be obtained by discretisation, i.e. such that
does not exist, where
and $\#$ denotes the cardinality of the set. I have two questions:
1) I have come up with the following example and I'd like to check whether it actually work, as I am not certain I understood the measure correctly.
$E := \{ (x,y) \in \mathbb{R}^2: y = \frac{1}{2},~x\in [0,1]\}$
If I understand the measure right, as $N = 1,2,3,4,5,6,7,\dots$, $m(E) = 0, \frac{3}{2},0 \frac{5}{4},0,\frac{7}{6},\dots$ hence the limit does not exist. Is this correct?
2) Do you know of simpler/ more enlightening examples?
Your idea is not correct, since your example is two-dimensional. The set $E\cap\frac{1}{N}\mathbb Z$ has $N+1$ elements if $N$ is even, and is empty if $N$ is odd. But then you devide by $N^2$, which gives you 0 for the limit in any case.
However, your example was not that bad :) Maybe we can use the following example for the one-dimensional case, which is a modification of yours. Define $$ E=\left\{\frac{a}{b}\in(0,1)\ |\ b\mbox{ is even}\right\} $$ as the set of all rationals between 0 and 1 with even denominator. Then $E\cap\frac{1}{N}\mathbb Z$ is empty iff $N$ is an odd prime, but has $N−1$ elements if $N$ is a power of 2. Thus, the limit cannot exist.