📂Measure Theory

Borel Set

Definition ¹

Let $\mathcal{F}$ be a sigma field of the Euclidean space $\mathbb{R}$.

We say that $\displaystyle \mathcal{B} : = \bigcap \left\{ \mathcal{F} : \mathcal{I} \subset \mathcal{F} \right\}$ is generated by the set $\mathcal{I}$ of all intervals. We call $B \in \mathcal{B}$ a borel set, and we call $\mathcal{B}$ the borel sigma field.

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• $\mathcal{I}$ is the set of all intervals.

Explanation

Simply put, it is the smallest sigma algebra among those that contain every interval. You can picture it as keeping everything that ought to be there while cutting out what is useless and leaving only what is necessary.

Contrary to its seemingly abstruse definition, borel sets have a wide range of uses, and they are especially useful when describing probability theory. By definition, a borel set appears as a union or intersection of intervals, examples of which include intervals, open sets, and countable sets. Here one must be able to read the true intent of the borel sigma field. A borel set is important not because it is literally an ‘interval’, but because it is a set built out of ‘open sets’ and ‘closed sets’.

By considering such borel sets, we become able to use various theorems of topology. In fact, even a mere topological space is general and abstract enough for the various application fields of measure theory. Once one accepts this discussion, the borel sigma field too can be accepted simply and easily as follows:

• The borel sigma field is just a restriction imposed in order to use topology.
• The borel sigma field is fairly small, so there is not much to consider.
• Hence in most theorems it is common to assume only the borel sigma field as a premise.

Intersection of a Set of Sets of Sets

Honestly, the definition of the borel sigma field is rather revolting. First, since an interval is a set, $\mathcal{I}$ is naturally a set of sets. But then, since we consider the set that gathers all sigma fields containing $\mathcal{I}$, it is a set of sets of sets. Taking the intersection there gives the borel sigma field. It is normal to find it hard to grasp.

Theorem

Let $\mathcal{M}$ be the sigma algebra that is the set of measurable sets of $X = \mathbb{R}$, that is, some sigma field containing every interval. Regarding this, the following hold.

• [1] The intersection of σ-fields is a σ-field, and therefore $\mathcal{B}$ is also a σ-field.
• [2]: $\mathcal{B}$ is the smallest σ-field containing every interval.
• [3]: $\mathcal{N} \subseteq \mathcal{B} \subsetneq \mathcal{M}$
• [4]: For every $ E \in \mathcal{M}$, there exists $B \in \mathcal{B}$ satisfying the following. $$ E \subset B \\ m(E ) = m(B) \\ m(B \setminus E) = 0 $$

Generalization

• General Borel Set