09 August 2019

I found an interesting exercise in Terry Tao’s Measure Theory book:

Proposition: for any $N \in \mathbb{R}$ and interval $I = [a, b]$ over $\mathbb{R}$, $|I| = \lim_{N \to \infty} \frac{|I ~ \cap ~ \frac{\mathbb{Z}}{N} |}{N},$ where $\frac{\mathbb{Z}}{N} := \{ \frac{n}{N} \mid n \in \mathbb{Z} \}$.

The interesting thing about this proposition is that it expresses the length of an interval $|I| = (b - a)$ (which is a continuous measure) in terms of the cardinality of a discrete set, $I \cap \frac{\mathbb{Z}}{N}$. To develop an intuition for how this proposition could work, first notice that $I \cap \frac{\mathbb{Z}}{N}$ is the set of points in $\frac{\mathbb{Z}}{N}$ which lie within the interval $I$. For example, if $a = 3$, $b = 8$, and $N = 25$, then $I \cap \frac{\mathbb{Z}}{N} = \{ {75}/{25}, {76}/{25}, \ldots, {200}/{25} \}$, which means its cardinality is $(200 - 75) + 1 = 126$.

Plugging this back into the right-hand side of the proposition, we see that:

As we can see, this is pretty close to $|I| = 5$. This example makes it clear that if the set $I \cap \frac{\mathbb{Z}}{N}$ can be expressed as $N(b - a) + \epsilon$ where $\epsilon$ is some constant, then the proposition pretty much holds at the limit. To get there, consider $\frac{l}{N}$ and $\frac{h}{N}$, the smallest and largest elements of the set ($l, h \in \mathbb{Z}$). Then, from our prior example, it is clear that the set cardinality can be expressed as:

Now, note that $\frac{l}{N} \ge a$ and $\frac{(l - 1)}{N} \lt a$, which can be rewritten as:

Similarly, we get:

Plugging these inequalities into the expression for the interval size, we get:

which we can rewrite as:

where $\epsilon > -1$ is a constant. Plugging this back into the right hand side of the original assertion, we get: