09 August 2019

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:

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