Properties of the Total Variation: a Finite Measure, the Jordan Decomposition, Linear Combinations, the Total Variation Norm, and Integrability

In the setting of Radon Measures on Locally Compact Metric Spaces: Standing Notation, let $(X,\mathcal{F})$ be a measurable space, let $\nu$ and $\nu'$ be finite signed measures on $(X,\mathcal{F})$, and let $c\in\mathbb{R}$. Let $|\nu|$ be the total variation and $\|\nu\|$ the total variation norm of $\nu$, and let $\nu^{+}$ and $\nu^{-}$ be its Jordan decomposition, formed with a Hahn set for $\nu$.

(i) A finite measure. $|\nu|$ is a finite measure on $(X,\mathcal{F})$.

(ii) Jordan decomposition. For every Hahn set $P$ for $\nu$ and every $A\in\mathcal{F}$,

$$|\nu|(A)=\nu^{+}(A)+\nu^{-}(A)=\nu(A\cap P)-\nu(A\setminus P),$$

and $\nu^{+}(A)\le|\nu|(A)$ and $\nu^{-}(A)\le|\nu|(A)$.

(iii) Bound. For every $A\in\mathcal{F}$,

$$|\nu(A)|\le|\nu|(A)\le\|\nu\|.$$

(iv) Nonnegative signed measures. Suppose that $\nu(A)\ge0$ for every $A\in\mathcal{F}$. Then $\nu$ is a finite measure on $(X,\mathcal{F})$, $X$ is a Hahn set for $\nu$, and $|\nu|(A)=\nu^{+}(A)=\nu(A)$ and $\nu^{-}(A)=0$ for every $A\in\mathcal{F}$. Moreover every $f\in\mathcal{L}^{1}(\nu)$ lies in $\mathcal{L}^{1}(\nu^{+})$ and in $\mathcal{L}^{1}(\nu^{-})$, and

$$\int f\,d\nu^{+}-\int f\,d\nu^{-}=\int f\,d\nu .$$

(v) Linear combinations. $\nu+\nu'$ and $c\nu$ are finite signed measures on $(X,\mathcal{F})$.

(vi) Total variation of linear combinations. For every $A\in\mathcal{F}$,

$$|\nu+\nu'|(A)\le|\nu|(A)+|\nu'|(A),\qquad|c\nu|(A)=|c|\,|\nu|(A).$$

(vii) The total variation norm. $M(X,\mathcal{F})$ is a real vector space under the setwise operations, with zero vector the zero function on $\mathcal{F}$, and $\nu\mapsto\|\nu\|$ is a norm on it, so that $(M(X,\mathcal{F}),\|\cdot\|)$ is a real normed space. In particular $\|\nu\|=0$ if and only if $\nu(A)=0$ for every $A\in\mathcal{F}$.

(viii) Integrability. Let $f:X\to\mathbb{R}$ be measurable. Then, for the integrals of $|f|\in\mathcal{M}^{+}$,

$$\int|f|\,d|\nu|=\int|f|\,d\nu^{+}+\int|f|\,d\nu^{-},$$

and $f\in\mathcal{L}^{1}(|\nu|)$ if and only if $f\in\mathcal{L}^{1}(\nu^{+})$ and $f\in\mathcal{L}^{1}(\nu^{-})$.