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The Cantor slab $C \times [0, 1]$ has content zero in $R^{2}$

Example

For the ordinary Cantor set $C$ , the slab $C \times [0, 1]$ has content zero in $R^{2}$ , and hence Jordan content $0$ .

Facts & Assumptions

Given: The Cantor set $C$ .

[L1]

$C$ has one-dimensional content zero (The Cantor set is compact, perfect, uncountable, nowhere dense and of measure zero, and it contains no interval of positive length, so its only nonempty connected subsets are single points, Measure zero (a countable cover by intervals of total length below every $ε$ ) and content zero (a finite such cover)).

[L2]

Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ controls the number of equal squares needed to stack across height $1$ .

Verification

technique · constructive

1.1

Given $ε > 0$ , cover $C$ by finitely many positive-width intervals $I_{r}$ with $\sum_{r} ℓ_{r}$ and $\max_{r} ℓ_{r}$ sufficiently small. Degenerate members may be enlarged within the budget.

L1chooseconstruct

1.2

Above $I_{r}$ , stack squares of side $ℓ_{r}$ . By [L2], at most $1 / ℓ_{r} + 2$ squares suffice, with total area at most $ℓ_{r} + 2 ℓ_{r}^{2}$ .

L2given

2.1

Summing gives at most $\sum_{r} ℓ_{r} + 2 (\max_{r} ℓ_{r}) \sum_{r} ℓ_{r} < ε$ . Thus the slab has cube-content zero.

step 1.1step 1.2given

3.1

By Jordan inner and outer content and Jordan measurable bounded sets in $R^{m}$ , cube-content zero makes the Jordan outer content $0$ . The nonnegative inner content is at most the outer content, so both are $0$ ; the slab is Jordan measurable with content $0$ , unlike the fat-Cantor slab The Smith–Volterra–Cantor slab $S \times [0, 1]$ is compact and not Jordan measurable.

step 2.1givendischarge-construct∎

Depends on

Used by

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Dependency tree · two levels

70 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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)

The Cantor slab C×[0,1]C×[0,1] has content zero in R2R2