Total Variation

• categories: Measure Theory, Definition, Functional analysis

The total variation of a signed measure provides a way to measure its “size” by converting it into a non-negative measure. It captures the “magnitude” of a signed measure without regard to sign

Definition

Let $(X,\Sigma )$ be a measurable space, and let $\mu$ be a signed measure on $(X,\Sigma )$. The total variation of $\mu$, denoted by $|\mu |$, is defined as:

$$|\mu |(A)=\sup \sum _{i=1}^n|\mu (A_i)|,$$

where the supremum is taken over all finite measurable partitions $\{A_i{\}}_{i=1}^n$ of $A$, with $A={\bigcup }_{i=1}^n{A}_i$ and $A_i\in \Sigma$ disjoint.

Properties

1. Non-Negativity: $|\mu |$ is a non-negative measure, even though $\mu$ itself may take negative values on certain sets.
2. Total Variation Norm: The total variation norm of $\mu$ is defined as $\Vert \mu {\Vert }_{TV}=|\mu |(X)$. This norm is used to quantify the “size” of $\mu$ and is fundamental in defining the space of signed measures.
3. Boundedness: For any measurable set $A\subset X$, we have $|\mu (A)|\le |\mu |(A)$.

Jordan Decomposition and Total Variation

The total variation $|\mu |$ of a signed measure $\mu$ can also be expressed in terms of the Jordan Decomposition $\mu ={\mu }^{+}-{\mu }^{-}$, where ${\mu }^{+}$ and ${\mu }^{-}$ are the positive and negative parts of $\mu$. In this case:

$$|\mu |={\mu }^{+}+{\mu }^{-}.$$

Intuition

The total variation $|\mu |$ measures the “absolute size” of a signed measure by summing up the absolute values of $\mu$ over disjoint sets. It effectively “removes” the sign from $\mu$, allowing us to treat it similarly to a non-negative measure, which simplifies analysis.

Applications

1. Norm on the Space of Signed Measures: The total variation norm $\Vert \mu {\Vert }_{TV}$ is used as the standard norm in the space of signed measures, making this space a complete metric space.
2. Absolute continuity: In the Radon-Nikodym Theorem, the total variation measure $|\mu |$ helps in defining when a signed measure is absolutely continuous with respect to another measure.
3. Convergence of Signed Measures: The total variation norm is used to define convergence criteria for sequences of signed measures, such as total variation convergence.