Signed measures and the Jordan decomposition

A finite signed measure assigns a real number to each measurable set and adds over countably many disjoint sets. It can be constructed as the difference of two finite positive measures. Conversely the decomposition theorem will show every finite signed measure has such a representation. Unlike a positive measure, it need not respect inclusion: adding a region with negative mass can reduce the total. Consequently its value on the whole space does not bound the size of its individual positive and negative contributions.

If h is integrable with respect to a positive measure μ, then ν(A)=∫A h dμ defines a finite signed measure. Integrability of |h| ensures countable additivity and prevents ambiguous infinite cancellation. The set A may contain both signs of h, so ν(A)=0 does not imply h vanishes there or that every subset has zero ν mass. Distinguishing a zero net value from a region on which the entire measure vanishes is essential when discussing positive sets and negative sets.

A positive set P for ν is a measurable set whose every measurable subset has nonnegative ν value. A negative set N has the corresponding nonpositive property. Hahn's theorem supplies a measurable partition X=P∪N into one positive and one negative set. It is stronger than merely asking ν(P)≥0 and ν(N)≤0: a positive net total could otherwise hide a negative subset. For a density h, the obvious partition uses h≥0 and h<0, with the zero region assignable to either side.

Define ν⁺(A)=ν(A∩P) and ν⁻(A)=−ν(A∩N). These are finite positive measures, they are mutually singular because they live on disjoint pieces, and their difference equals ν. Hahn partitions may differ on regions carrying no positive or negative mass, but the two measures ν⁺ and ν⁻ are independent of that choice. Their uniqueness is the Jordan decomposition theorem. An arbitrary difference of positive measures is not necessarily Jordan: shared mass can be present on both sides until cancellation is removed.

The variation of ν on A can also be defined as the supremum, over finite measurable partitions of A, of the sum of the absolute ν values of the pieces. Refining a partition exposes cancellations hidden inside larger pieces. For a density h, variation has density |h|, while the Jordan parts have densities max(h,0) and max(−h,0). This connects the abstract decomposition directly to positive and negative parts of integrable functions. It also gives the bound |ν(A)|≤|ν|(A) for every measurable A.

Integration against ν is defined through its Jordan parts, with absolute integrability measured using |ν|. For a suitable f, the magnitude of ∫f dν is at most ∫|f| d|ν|. This is useful when comparing two probability measures: their signed difference has total mass zero, but its variation quantifies disagreement on events. Numerical conventions for total variation distance may include a factor one half, so the variation measure and a normalized probability distance should be named separately rather than identified silently.