SUMMARY
The Riemann integral of a function defined on a closed Jordan domain within the set ##\pi=\{ x\in\mathbb{R}^n \;|\; x=(x_1,...,x_{n-1}, 0) \}## is zero if the function is Riemann integrable. Specifically, for any closed Jordan domain ##E \subset \pi## and a Riemann integrable function ##f: E \rightarrow \mathbb{R}##, the integral satisfies ##\int_{E} f(x) dV = 0##. The proof involves demonstrating that for any given ##\varepsilon > 0##, there exist upper and lower sums ##U## and ##L## such that ##-\varepsilon < L < U < \varepsilon##.
PREREQUISITES
- Understanding of Riemann integrability and its criteria.
- Familiarity with closed Jordan domains in mathematical analysis.
- Knowledge of upper and lower sums in the context of integration.
- Basic concepts of measure theory related to sets in ##\mathbb{R}^n##.
NEXT STEPS
- Study the properties of closed Jordan domains in detail.
- Learn about the criteria for Riemann integrability and its implications.
- Explore the concept of upper and lower sums in Riemann integration.
- Investigate measure theory fundamentals, particularly in ##\mathbb{R}^n##.
USEFUL FOR
Mathematics students, educators, and researchers focusing on real analysis, particularly those studying integration theory and its applications in higher dimensions.