Strengthening the non-density conclusion to finite measure for Lipschitz images on chain-connected sequentially compact metric spaces

Determine whether, for every sequentially compact and chain-connected metric space (M, d) and every Lipschitz continuous function f: M → ℝ, the image f(M) necessarily has finite Lebesgue measure. This asks whether item (h2) in Theorem \ref{preppiu}—which establishes that f(M) is not dense in ℝ—can be strengthened to the statement that f(M) has finite measure.

Background

Theorem \ref{preppiu} establishes several properties of continuous and Lipschitz functions on sequentially compact, chain-connected metric spaces, including a version of the intermediate value theorem and consequences for the image of such functions.

Item (h2) in Theorem \ref{preppiu} proves that for Lipschitz f: M → ℝ under these structural conditions on (M, d), f(M) is not dense in ℝ. The authors explicitly state they could not strengthen this to a finite measure conclusion, raising the question whether a stronger quantitative measure-theoretic property holds.

Resolving this would clarify the measure-theoretic structure of images of Lipschitz maps under the connectedness and compactness assumptions used throughout the paper.