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The Cantor slab has content zero in
Example
For the ordinary Cantor set , the slab has content zero in , and hence Jordan content .
Facts & Assumptions
Given: The Cantor set .
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 ) and content zero (a finite such cover)).
Integer part: for every real there is exactly one integer with controls the number of equal squares needed to stack across height .
Verification
Given , cover by finitely many positive-width intervals with and sufficiently small. Degenerate members may be enlarged within the budget.
Above , stack squares of side . By [L2], at most squares suffice, with total area at most .
Summing gives at most . Thus the slab has cube-content zero.
By Jordan inner and outer content and Jordan measurable bounded sets in , cube-content zero makes the Jordan outer content . The nonnegative inner content is at most the outer content, so both are ; the slab is Jordan measurable with content , unlike the fat-Cantor slab The Smith–Volterra–Cantor slab is compact and not Jordan measurable.
Depends on
- 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 $\varepsilon$) and content zero (a finite such cover)
- Jordan inner and outer content and Jordan measurable bounded sets in $\mathbb{R}^m$
- Axis-parallel rectangles in $\mathbb{R}^m$ and their volume
- Integer part: for every real $x$ there is exactly one integer $m$ with $m \le x < m + 1$
- The canonical natural $\iota(n) = n \cdot 1_F$ of a field
- The Smith–Volterra–Cantor slab $S\times[0,1]$ is compact and not Jordan measurable
Used by
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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)