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

Proposition: for any and interval over , where .

The interesting thing about this proposition is that it expresses the length of
an interval (which is a *continuous* measure) in terms of the
cardinality of a *discrete* set, .
To develop an intuition for how this proposition could work, first notice that
is the set of points in
which lie within the interval .
For example, if , , and , then
,
which means its cardinality is
.

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

As we can see, this is pretty close to .
This example makes it clear that
if the set can be expressed as
where is some constant,
then the proposition pretty much holds *at the limit*.
To get there, consider and , the smallest and
largest elements of the set ().
Then, from our prior example, it is clear that the set cardinality can be
expressed as:

Now, note that and , 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 is a constant. Plugging this back into the right hand side of the original assertion, we get: