Interactive Real Analysis

• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 7.1. Riemann Integral
• 7.2. Integration Techniques
• 7.3. Measures
• 7.4. Lebesgue Integral
• 7.5. Riemann versus Lebesgue
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

7.3. Measures

Theorem 7.3.8: Borel Sets are Measurable

The collection of Borel sets is the smallest sigma-algebra which contains all of the open sets. Every Borel set, in particular every open and closed set, is measurable.

Context

This follows from the fact that open sets and closed sets are measurable, as we have just proved, and so are countable unions (and intersections) of those sets. Therefore the collection of all measurable sets is a sigma-algebra. But then, since by definition the Borel sets are the smallest sigma algebra containing the open sets, it follows that the Borel sets are a subset of all measurable sets and are therefore measurable.