Signed Measure - Properties

Properties

What follows are two results which will imply that an extended signed measure is the difference of two nonnegative measures, and a finite signed measure is the difference of two finite non-negative measures.

The Hahn decomposition theorem states that given a signed measure μ, there exist two measurable sets P and N such that:

1. P∪N = X and P∩N = ∅;
2. μ(E) ≥ 0 for each E in Σ such that E ⊆ P — in other words, P is a positive set;
3. μ(E) ≤ 0 for each E in Σ such that E ⊆ N — that is, N is a negative set.

Moreover, this decomposition is unique up to adding to/subtracting μ-null sets from P and N.

Consider then two nonnegative measures μ+ and μ- defined by

and

for all measurable sets E, that is, E in Σ.

One can check that both μ+ and μ- are nonnegative measures, with one taking only finite values, and are called the positive part and negative part of μ, respectively. One has that μ = μ+ - μ-. The measure |μ| = μ+ + μ- is called the variation of μ, and its maximum possible value, ||μ|| = |μ|(X), is called the total variation of μ.

This consequence of the Hahn decomposition theorem is called the Jordan decomposition. The measures μ+, μ- and |μ| are independent of the choice of P and N in the Hahn decomposition theorem.