Zeroth canonical homomorphism from singular to Milnor-Thurston homology is injective

Milnor-Thurston homology also known as “measure homology” is a generalisation of singular homology where chains are defined to be measures (possibly supported on infinitely many singular simplices). In this paper we prove that the canonical homomorphism from singular to Milnor-Thurston homology is injective in dimension zero for metrizable spaces.