CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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$((0,1)\setminus S)\times(0,1)$ is bounded and open, but its boundary has positive Jordan outer content

Statement refuted

Every bounded open subset of $\mathbb R^2$ is Jordan measurable.

Facts & Assumptions

Given: $U=((0,1)\setminus S)\times(0,1)$ , with $S$ the fat Cantor set.

[L1]

$S$ is closed, nowhere dense, and has interval-cover lower bound $1/2$ (The Smith-Volterra-Cantor set: the same construction removing, at stage $n \ge 1$ , an open middle interval of length $4^{-n}$ from each of the $2^{n-1}$ remaining intervals, The Smith-Volterra-Cantor set is compact, perfect and nowhere dense, and does not have measure zero).

[L2]

The slab $S\times[0,1]$ has outer content at least $1/2$ (If every finite interval cover of $A\subseteq\mathbb{R}$ has total length at least $c$ , then every rectangle cover of $A\times[0,d]$ has total area at least $cd$ , The Smith–Volterra–Cantor slab $S\times[0,1]$ is compact and not Jordan measurable).

Counterexample

technique · direct

1.1

The set $(0,1)\setminus S$ is open, so $U$ is open and bounded.

1.2

Every neighbourhood of a point of $S\times[0,1]$ meets $U$ , by nowhere density in the first coordinate and the interval factor, and meets the complement. Hence $S\times[0,1]\subseteq\partial U$ .

L1given

2.1

Outer content is monotone under inclusion, so [L2] gives the boundary positive outer content.

step 1.2L2given

3.1

By A bounded set in $\mathbb{R}^m$ is Jordan measurable iff its boundary is null, equivalently of content zero, $U$ is not Jordan measurable.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 171 results over 25 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Basic Analysis, Jordan Measurable Sets (standard reference, not scraped)
J. Lebl, Basic Analysis, Outer Measure and Null Sets (standard reference, not scraped)
Whitman College real analysis notes (standard reference, not scraped)

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