signed measure

An ordinary measure assigns to each set a non-negative ‘amount’ — a size, a mass, a probability. A signed measure relaxes the non-negativity, allowing the amount to be negative. Picture a distribution of electric charge: some regions carry positive charge, others negative, and the ‘measure’ of a region is its net charge, which can come out either sign. This is the natural object for describing differences of measures.

Formally, a signed measure on a sigma-algebra is a set function nu that is countably additive — nu of a disjoint union equals the sum of the nu's — and takes values in the extended reals, with the rule that at most one of +infinity and -infinity is permitted as a value (to avoid the undefined infinity-minus-infinity). The empty set has measure 0. Every signed measure can be written as a difference nu = nu+ minus nu- of two ordinary (non-negative) measures with disjoint supports; this is the Jordan decomposition, and the underlying split of the space is the Hahn decomposition.

The total variation |nu| = nu+ plus nu- is a genuine non-negative measure that records the total ‘gross’ size, ignoring cancellation. A leading source of signed measures is integration: if g is integrable against a measure mu, then nu(E) = the integral of g over E defines a signed measure, and the Radon–Nikodym theorem says, conversely, that every signed measure absolutely continuous with respect to mu arises this way.