For two classes of non-CW spaces with countable fundamental group, the canonical map from singular homology with real coefficients to measure homology is injective in degrees at least 2.
The uncountability of the Harmonic Archipelago Group
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On measure homology of mildly wild spaces
For two classes of non-CW spaces with countable fundamental group, the canonical map from singular homology with real coefficients to measure homology is injective in degrees at least 2.