LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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.

Support of a continuous pushforward is the image closure

Statement

Let $X \subseteq R^{k}$ and $Y \subseteq R^{d}$ be closed Euclidean subspaces. Let $μ$ be a finite nonnegative Borel measure on $X$ , define $spt μ = {x : μ (U) > 0 for every relatively open neighborhood U of x}$ , and let $f : X \to Y$ be continuous. Define the pushforward by $f_{#} μ (A) = μ (f^{- 1} (A))$ for Borel $A$ . Then

$spt (f_{#} μ) = \overline{f (spt μ)} .$

Statement references

Chizat and Bach, Lemma C.8

Ambrosio, Gigli, and Savaré, Chapter 8

Facts & Assumptions

Given: $μ$ and $f$ as in the Statement.

Proof

1.1givenconstruct

If $y = f (x)$ for $x \in spt μ$ and $V$ is any open neighborhood of $y$ , continuity makes $f^{- 1} (V)$ an open neighborhood of $x$ . Hence $f_{#} μ (V) = μ (f^{- 1} (V)) > 0$ , so $y \in spt (f_{#} μ)$ . Taking the closure relative to $Y$ , since support is closed, $\overline{f (spt μ)} \subseteq spt (f_{#} μ)$ .

2.1givenstep 1.1choose∎

The complement of $spt μ$ has $μ$ -measure zero: every point in it lies in a zero-measure open neighborhood, and the countable relative Euclidean basis gives a countable cover by zero-measure basis sets. If $y \notin \overline{f (spt μ)}$ , choose an open neighborhood $V$ disjoint from that closure. Then $f^{- 1} (V) \subseteq X ∖ spt μ$ , so $f_{#} μ (V) = 0$ and $y \notin spt (f_{#} μ)$ . Thus $spt (f_{#} μ) \subseteq \overline{f (spt μ)}$ .

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Lénaïc Chizat and Francis Bach, On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport (standard reference, not scraped)
Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures (standard reference, not scraped)