The total variation of a signed charge (or signed measure) mu on an algebra A of subsets of X is defined by
V_mu = sup{sum_{i=1}^n |mu(A_i)| : {A_1, …, A_n} is a partition of X in A}.
A signed charge is of bounded variation if V_mu < infinity. The total variation defines an L-norm (lattice norm) on the space ba(A) of all signed charges of bounded variation, known as the total variation norm: ||mu|| = V_mu = |mu|(X), where |mu| is the total variation measure.
The total variation measure |mu| of a signed charge mu in ba(A) is characterised by
|mu|(A) = sup{sum_{i=1}^n |mu(A_i)| : {A_1, …, A_n} is a partition of A}.
This equals mu^+(A) + mu^-(A), where mu^+ and mu^- are the positive and negative parts from the Hahn-Jordan decomposition.
Total variation plays a central role in measure convergence for signed measures: tightness is defined in terms of the total variation measure (sup_n |mu_n|(K^c) ⇐ eps), the hierarchy between vague convergence, loose convergence, and weak convergence of measures is governed by local vs global boundedness in total variation, and the mass preserving condition hierarchy controls mass loss through the total variation of the Hahn-Jordan parts.
Key Details
- Norm: ||mu|| = V_mu = |mu|(X) = mu^+(X) + mu^-(X) on ba(A)
- Lattice: ba(A) with total variation norm is an AL-space (Abstract Lebesgue space)
- Completeness: ba(A) is a Banach lattice under the total variation norm
- For measures: ca(A) (countably additive signed measures) is a closed sublattice (projection band) of ba(A)
- Role in convergence: tightness uses |mu_n|; loose convergence requires uniform boundedness sup_n ||mu_n|| < infinity; vague convergence requires only local boundedness
Textbook References
Infinite Dimensional Analysis (Aliprantis-Border, 2006)
- Definition (p. 396): Total variation V_mu = sup{sum |mu(A_i)| : partition of X}; bounded variation iff V_mu < infinity
- Theorem 10.53 (p. 397): ba(A) is an AL-space with total variation norm; lattice operations given explicitly
- Corollary 10.54 (p. 398): mu^+(A) = sup{mu(B) : B subset A}; mu^-(A) = -inf{mu(B) : B subset A}; |mu|(A) = sup{sum |mu(A_i)| : partition of A}
- Corollary 10.55 (p. 399): A signed charge has bounded variation iff it has bounded range
- Theorem 10.56 (p. 399): ca(A) is a projection band in ba(A), also an AL-space with total variation norm