The paper investigates a
homology theory based on the ideas of Milnor and Thurston that by considering
measures on the set of all singular simplices one should get alternate possibilities for
describing the cycles of classical homology theory. It suggests slight changes to
Milnor’s and Thurston’s original definitions (giving differences for wild topological
spaces only) which ensure that their homology theory is well-defined on all
topological spaces. 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
these both homology theories does not hold for all topological spaces is also
included.
