On the (non)-coincidence of Milnor-Thurston homology theory with singular homology theory

The paper investigates a homology theory based on the ideas of Milnor and T hurston that by considering measures on the set of all singular simplices o ne should get alternate possibilities for describing the cycles of classica l homology theory. It suggests slight changes to Milnor's and Thurston's or iginal definitions (giving differences for wild topological spaces only) wh ich ensure that their homology theory is well-defined on all topological sp aces. It further proves that Milnor-Thurston homology theory gives the same homology groups as the singular homology theory with real coefficients for all triangulable spaces. An example showing that the coincidence between t hese both homology theories does not hold for all topological spaces is als o included.