Alphabeta Math

The Riemann Integral in R^m and Jordan Content: Examples and Counterexamples

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1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

The unit box in ℝᵐ has volume 1, and the integral of a constant c over it is c

Counterexample

The rational points of [0,1]² form a bounded null set that is not Jordan measurable

Counterexample

The Smith–Volterra–Cantor slab S×[0,1] is compact and not Jordan measurable

Counterexample

((0,1)∖ S)×(0,1) is bounded and open, but its boundary has positive Jordan outer content

Example

The parabola segment {(x,x²):0≤ x≤1} has content zero in ℝ²

Example

The right triangle {(x,y)∈[0,1]²:x+y≤1} has Jordan content 1/2

Example

The Cantor slab C×[0,1] has content zero in ℝ²

Remark

The term “rectifiable” for Jordan measurable sets is unrelated to rectifiable curves

Sources