This problem is still far from being solved.
Its description is far from complete.
To go into this in detail would take us too far afield.
However, (1) has been proved so far only in the case where......
So far we have not topologized M(R).
So far it seems not to be known whether the geometric condition on X can be omitted.
Functions which are equal almost everywhere are indistinguishable as far as integration is concerned.