The Lebesgue decomposition theorem for arbitrary contents (Q865018)

The decomposition theorem named after Lebesgue asserts that certain set functions have canonical representations as sums of absolutely continuous and singular set functions with respect to some fixed set function. The traditional versions are for bounded measures with respect to some fixed measure on a \(\sigma\)-algebra. In subsequent decades since Hahn's final form (1921), there have been several versions with particular requirements on the two constituents, as well as abstract extensions of particular versions. The author cites a list of these in the abstract of the paper. In an effort to unify the theory (find a ``common roof'' for partial notions), the present article claims to arrive at the aim in the original context of arbitrary contents, using the difference formation introduced by the author in his 1999 paper. This difference formation also affords explicit formulas for the two constituents.