1. What Is a Measure?

Imagine you’re trying to figure out how to assign a size, or “measure,” to different kinds of sets—some simple, like intervals on the real line, and some more complex. A measure is just a mathematical way to do this, and it needs to follow a few basic rules:

  1. Non-negativity: The measure of any set should be non-negative (you can’t have negative length or size).
  2. The empty set gets zero: The set with nothing in it has a measure of zero.
  3. Additivity: If you can break up a set into disjoint pieces, then the measure of the whole set should be the sum of the measures of the pieces.

A probability measure is a special kind of measure where the total size of the space is 1—think of it like spreading probability evenly across all outcomes in a probability space.

2. The Lebesgue Measure on $ [0, 1] $

The unit interval $ [0, 1] $ is foundational to probability theory, often used to parameterize probabilities. In addition, one of the most common measures is the Lebesgue measure, which is a natural way to assign “length” to sets in $ \mathbb{R} $. On the unit interval $ [0, 1] $, this measure behaves exactly as you’d expect: the measure of an interval is just its length. For example:

$$ \lambda([a, b]) = b - a $$

where $ a \leq b $ are points in $ [0, 1] $.

This works perfectly well for many sets. For example, if you take disjoint intervals or measurable subsets of $ [0, 1] $, the Lebesgue measure adds up in a way that makes sense. But things get tricky when we push this idea to its limits, especially with more exotic sets.

3. What Happens with Countable Partitions?

Here’s where things get weird. Let’s say we try to break $ [0, 1] $ into an infinite number of disjoint sets. According to the rules of a measure, the measure of the whole interval should be the sum of the measures of each piece.

But if each set in the partition has a small positive measure, we run into trouble. The sum of infinitely many positive numbers would blow up, exceeding the measure of $ [0, 1] $, which is just 1. On the other hand, if each set has measure zero, then summing them all would still give zero, contradicting the fact that the total measure of $ [0, 1] $ is 1. This shows that trying to break $ [0, 1] $ into a countable collection of measurable sets doesn’t always work cleanly.

4. The Vitali Set: A Pathological Example

Let’s get even more specific. The Vitali set gives a famous example of something that can’t be measured using the Lebesgue measure. Here’s how it works:

  1. Equivalence by Rational Shifts: We say two points $ x $ and $ y $ in $ [0, 1] $ are equivalent if their difference is a rational number, $ x - y \in \mathbb{Q} $. This partitions $ [0, 1] $ into equivalence classes.
  2. Constructing the Vitali Set: From each equivalence class, we pick exactly one representative to form the Vitali set $ V $. No two points in $ V $ are rationally related.

Here’s where it gets strange: by shifting $ V $ by every rational number in $ [0, 1] $ (cyclically wrapping around using mod 1), you can essentially reconstruct the entire interval $ [0, 1] $.

5. Why the Vitali Set Can’t Be Measured

At first glance, it seems like this set should have a well-defined measure. But here’s the catch: let’s try to assign a measure to it.

If each shifted version of $ V $ has the same measure, we could write:

$$ [0,1] = \bigcup_{q \in \mathbb{Q} \cap [0,1]} (V + q \mod 1) $$

So the measure of the entire interval should be the sum of the measures of all these disjoint shifted sets. But since there are infinitely many shifts (because the rationals in $ [0,1] $ are countable but infinite), we face the same contradiction as before: either the measure is zero, which contradicts the fact that the measure of $ [0, 1] $ is 1, or it’s positive and infinite, which doesn’t make sense either.

This shows that the Vitali set is non-measurable—it’s not something the Lebesgue measure can handle.

The construction of the Vitali set hinges on the Axiom of Choice, a fundamental principle in set theory. This axiom allows us to select exactly one representative from each equivalence class, even when no explicit rule dictates how to make these selections. Without the Axiom of Choice, constructing such a set would not be guaranteed.

This is important because the Vitali set cannot be measured using the Lebesgue measure. Its construction shows how the Axiom of Choice can lead to the existence of non-measurable sets.

Borel Sets and the Borel Measure

1. Borel Sets: A Well-Behaved Family

So what do we do to avoid these kinds of paradoxes? We focus on sets that behave nicely. A Borel set is any set you can build from open sets using countable unions, intersections, and complements. These sets form the Borel σ-algebra, denoted $ \mathcal{B}(\mathbb{R}) $. This family of sets includes familiar things like intervals, points, and other standard subsets of the real line.

The important part is that Borel sets avoid pathological examples like the Vitali set, so they play well with our idea of what a measure should do.

2. The Borel Measure

The Borel measure is simply a measure defined on Borel sets. It inherits all the good properties of the Lebesgue measure when restricted to these sets, ensuring that we don’t run into the problems we saw earlier.

For any Borel set $ A $, the Borel measure $ \mu $ satisfies:

  1. Non-negativity: $ \mu(A) \geq 0 $.
  2. Countable Additivity: If $ {A_i} $ is a countable collection of disjoint Borel sets, then $ \mu(\bigcup A_i) = \sum \mu(A_i) $.
  3. Length Consistency: The measure of an interval matches our intuitive notion of length: $$ \mu([a, b]) = b - a $$ This makes the Borel measure a more controlled version of the Lebesgue measure.

3. Borel Probability Measure

In many applications, especially in probability theory, we want the total measure to be 1. A Borel probability measure is just a Borel measure that assigns total measure 1 to the space. For the unit interval $ [0,1] $, we define $ P([0,1]) = 1 $, and this measure is perfect for modeling probabilities tied to intervals or more complex sets in a way that avoids the problems posed by non-measurable sets.

4. Why the Borel Measure Works

By restricting ourselves to Borel sets, we sidestep the paradoxes introduced by non-measurable sets like the Vitali set. The Borel measure is finite on intervals, respects countable additivity, and remains consistent with probability theory and integration.

Conclusion

The Borel measure strikes a balance between flexibility and control. By focusing on Borel sets, we keep the familiar and intuitive properties of the Lebesgue measure, while avoiding paradoxes that arise when we try to measure more exotic sets. It’s a clean, consistent framework that ensures we can handle most sets that arise in real analysis and probability theory without running into contradictions.