Borel Sigma-Algebra

Compact study note.

Summary

The Borel sigma-algebra on the real line is the smallest sigma-algebra containing all open subsets of $\mathbb{R}$ . It is the standard measurable-event structure for real-valued random variables.^[1]

Prerequisites

• Sigma-Algebra

Notation and Assumptions

$\mathcal{B}(\mathbb{R})=\sigma (\{G\subseteq \mathbb{R}:G\text{ is open}\})$ . It also contains closed intervals, half-open intervals, countable sets including $\mathbb{Q}$ , and the set of irrational numbers $\mathbb{R}\setminus \mathbb{Q}$ .

Essential Result

One real-valued random variable $X$ is measurable when $X^{-1}(B)\in \mathcal{F}$ for every $B\in \mathcal{B}(\mathbb{R})$ . Not every Borel set is simply 'closed minus open'; the correct definition is generation by open sets.

Small Example

Closed intervals are Borel. For example,

Common Mistakes

• Claiming every Borel set has one simple closed-minus-open form.
• Using nonstandard phrases including 'irrational line' instead of 'set of irrational numbers'.

Connections

• Sigma-Algebra
• Random Variable
• Distribution Induced by a Random Variable

References