LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite volume and interior density force exact flow invariance

Statement

Let $C$ be a connected manifold with a Borel measure $μ$ positive on every nonempty open set. Let $ψ_{s}$ be a complete flow by measure-preserving homeomorphisms. Let $B \subset C$ be closed, $μ (B) < \infty$ , $B = {\overline{{int}_{C} B}}^{C}$ , and $ψ_{s} (B) \subset {int}_{C} B$ for every $s > 0$ . Then $ψ_{s} (B) = B$ for every $s > 0$ , and $B$ is open as well as closed. If $C ∖ B$ is nonempty, then $B = \emptyset$ .

Facts & Assumptions

[F1]

In its stated geometric setting, the inaccessible-set construction supplies the relative-interior and closure properties assumed here. Inaccessible set is the closure of its relative interior

[F2]

Under the shift-flow lemma's hypotheses, its complete volume-preserving diffeomorphism flow and positive Riemannian Borel measure supply instances of the flow and measure data assumed here. Complete volume preserving shift flow on a spherical exterior

[F3]

The manifold $C$ carries its topology. Finite intersections of open sets are open, and a set is closed precisely when its complement is open. Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison

[F4]

The Borel sigma-algebra is generated by the open sets and is closed under complements and countable unions, hence also under differences. A Borel measure is a nonnegative, countably additive function on this sigma-algebra with $μ (\emptyset) = 0$ . The Borel sigma-algebra of a topological space Sigma-algebras Measures on sigma-algebras

[F5]

Here a complete flow by homeomorphisms means a continuous map $ψ : R \times C \to C$ , with $ψ_{0} = {id}_{C}$ and $ψ_{s + t} = ψ_{s} \circ ψ_{t}$ , each $ψ_{s}$ being a continuous bijection with continuous inverse $ψ_{- s}$ . Measure preservation means $μ (ψ_{s}^{- 1} (E)) = μ (E)$ for every Borel set $E$ , where $ψ_{s}^{- 1} (E)$ denotes inverse image. Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological Measure-preserving transformations and systems

[F6]

Interior is the largest open subset, and closure is the smallest closed superset, in the topology of $C$ . A point belongs to the closure of a set exactly when every open neighborhood of that point meets the set. Interior, closure, boundary, exterior, derived set and isolated point in a topological space A point lies in the closure of A iff every basic neighbourhood of it meets A; the closure is the smallest closed superset and equals A together with its derived set

[F7]

Connectedness means that there is no partition of $C$ into two nonempty disjoint open sets. Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets

Proof

Given: The hypotheses in the Statement.

Goal: Establish exact invariance, openness and the stated empty-set conclusion.

1.1givenF3F4F5

Fix $s > 0$ and put $A = ψ_{s} (B)$ . Since $ψ_{s}$ is a homeomorphism, $A$ is closed: its complement is the inverse image of the open set $C ∖ B$ under the continuous inverse map $ψ_{s}^{- 1}$ . Thus $A$ and $B$ are Borel. Measure preservation applied to $A$ gives $μ (A) = μ (ψ_{s}^{- 1} (A)) = μ (B) < \infty$ , since bijectivity gives $ψ_{s}^{- 1} (A) = B$ .

2.1givenF4F6step 1.1

The hypothesis gives $A \subset {int}_{C} B \subset B$ . The difference $B ∖ A$ is Borel. Apply countable additivity to the disjoint sets $A$ , $B ∖ A$ , followed by empty sets: $μ (B) = μ (A) + μ (B ∖ A)$ . The equality of the first two finite measures from step 1.1 yields $μ (B ∖ A) = 0$ .

3.1givenF3F4F6step 1.1step 2.1

Suppose $x \in B ∖ A$ . The set $O = C ∖ A$ is open and contains $x$ . Since $B = {\overline{{int}_{C} B}}^{C}$ , the closure characterization gives $V = O \cap {int}_{C} B \neq \emptyset$ . The set $V$ is open, so $μ (V) > 0$ , and $V \subset B ∖ A$ . Both $V$ and $(B ∖ A) ∖ V$ are Borel; finite additivity gives $0 = μ (B ∖ A) = μ (V) + μ ((B ∖ A) ∖ V) \geq μ (V) > 0$ , a contradiction. Therefore $ψ_{s} (B) = B$ , as an exact set identity, for every $s > 0$ .

4.1givenF3F6F7step 3.1∎

Fix any $s > 0$ . Equality from step 3.1 and the assumed forward inclusion give $B = ψ_{s} (B) \subset {int}_{C} B \subset B$ , so $B$ is open as well as closed. If $C ∖ B$ is nonempty and $B$ were nonempty, the two complementary open sets $B$ and $C ∖ B$ would be a separation of $C$ . Connectedness rules this out, so $B = \emptyset$ .

Depends on

Used by · 1

A maximal stationary exterior has future access to its far tail

Dependency tree · 2 levels

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