Interactive Real Analysis - part of MathCS.org

• 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

Example 7.4.7(c): Riemann implies Lebesgue Integrable

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If possible, find the Riemann and Lebesgue integrals of the constant function f(x) = 1 over the Cantor middle-third set.

Context

It does not make sense to ask for the Riemann integral of a function defined over a set that is not an interval, so there's nothing to do for the second part.

But we could rephrase the question: take the function f that is equal to 1 over the Cantor set and zero everywhere else and find the Riemann integral of that function over some interval. Now it is a well-defined question, but the Riemann integral does not exist, which is - again (!) - left as an exercise.