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A bounded set in is Jordan measurable iff its boundary is null, equivalently of content zero
Statement
A metric-bounded set is Jordan measurable if and only if its boundary is null, equivalently has content zero.
Facts & Assumptions
Given: Metric-bounded in the sense of Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space.
If is a nondegenerate rectangle with , then the relative-domain indicator is discontinuous exactly at the ambient boundary . At a boundary point every sufficiently small ambient ball lies in and meets both and its ambient complement, while away from the boundary the indicator is locally constant (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space).
Indicator integrability is Jordan measurability (A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content), and integrability is equivalent to a null discontinuity set (Lebesgue's criterion in : a bounded function on a closed nondegenerate rectangle is Riemann integrable iff its discontinuity set is null).
The boundary is closed: it is the intersection of the closed set with the complement of the open set (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space, The closure of a nonempty is , equals together with its limit points, and is the smallest closed superset, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement). By Jordan inner and outer content and Jordan measurable bounded sets in , metric boundedness places in a closed bounding rectangle . Since is closed and contains , the smallest-closed-superset property gives . Thus is closed and bounded, hence compact by Heine-Borel in : with the Euclidean metric a subset of is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, and compact nullity is equivalent to content zero (For compact subsets of , measure zero and content zero coincide).
Proof
By [L3], choose a closed bounding rectangle for and enlarge every coordinate interval by a fixed positive margin to obtain a nondegenerate rectangle with . By [L1] and [L2], is Jordan measurable exactly when is null.
By [L3], nullity of this compact boundary is equivalent to content zero.
Combining the equivalences proves the criterion.
Depends on
A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content
Lebesgue's criterion in : a bounded function on a closed nondegenerate rectangle is Riemann integrable iff its discontinuity set is null
Interior, closure, boundary, limit point, isolated point and dense subset of a metric space
The closure of a nonempty is , equals together with its limit points, and is the smallest closed superset
The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space
Heine-Borel in : with the Euclidean metric a subset of is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
For compact subsets of , measure zero and content zero coincide
Measure zero and content zero in by countable and finite cube covers
Jordan inner and outer content and Jordan measurable bounded sets in