Properties of the Integral with Respect to a Finite Signed Measure: Linearity, Bounds, Bounded Functions, Linearity in the Measure, Indicators and Finite Measures

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 $|\nu|$ and $\|\nu\|$ be the total variation and the total variation norm of $\nu$, and let $\mathcal{L}^{1}(\nu)$ and $\int f\,d\nu$ be as in Integrable Functions and the Integral with Respect to a Finite Signed Measure (i) and Integrable Functions and the Integral with Respect to a Finite Signed Measure (ii).

(i) Linearity. For $f,g\in\mathcal{L}^{1}(\nu)$ and $a,b\in\mathbb{R}$, $af+bg\in\mathcal{L}^{1}(\nu)$ and

$$\int(af+bg)\,d\nu=a\int f\,d\nu+b\int g\,d\nu .$$

(ii) Bound. For $f\in\mathcal{L}^{1}(\nu)$, $|f|\in\mathcal{L}^{1}(|\nu|)$ and

$$\Bigl|\int f\,d\nu\Bigr|\le\int|f|\,d|\nu| .$$

(iii) Bounded functions. Let $f:X\to\mathbb{R}$ be measurable and let $C\in\mathbb{R}$ with $C\ge0$ and $|f(x)|\le C$ for every $x\in X$. Then $f\in\mathcal{L}^{1}(\nu)$ and

$$\Bigl|\int f\,d\nu\Bigr|\le C\,\|\nu\| .$$

(iv) Differences of finite measures. Let $\mu_{1}$ and $\mu_{2}$ be finite measures on $(X,\mathcal{F})$ with $\nu(A)=\mu_{1}(A)-\mu_{2}(A)$ for every $A\in\mathcal{F}$, and let $f:X\to\mathbb{R}$ be measurable with $|f(x)|\le C$ for every $x\in X$, for some $C\in\mathbb{R}$ with $C\ge0$. Then

$$\int f\,d\nu=\int f\,d\mu_{1}-\int f\,d\mu_{2}.$$

(v) Linearity in the measure. Let $f:X\to\mathbb{R}$ be measurable with $|f(x)|\le C$ for every $x\in X$, for some $C\in\mathbb{R}$ with $C\ge0$. Then, for the finite signed measures $\nu+\nu'$ and $c\nu$ of Properties of the Total Variation: a Finite Measure, the Jordan Decomposition, Linear Combinations, the Total Variation Norm, and Integrability (v),

$$\int f\,d(\nu+\nu')=\int f\,d\nu+\int f\,d\nu',\qquad\int f\,d(c\nu)=c\int f\,d\nu .$$

(vi) Indicators. For every $A\in\mathcal{F}$, $\mathbf{1}_{A}\in\mathcal{L}^{1}(\nu)$ and

$$\int\mathbf{1}_{A}\,d\nu=\nu(A).$$

(vii) Finite measures. Suppose that $\nu(A)\ge0$ for every $A\in\mathcal{F}$. Then $\nu$ is a finite measure, the set $\mathcal{L}^{1}(\nu)$ of Integrable Functions and the Integral with Respect to a Finite Signed Measure (i) is the set $\mathcal{L}^{1}(\nu)$ of the base, and for every $f$ in it the integral of Integrable Functions and the Integral with Respect to a Finite Signed Measure (ii) equals the Lebesgue integral $\int f\,d\nu$ of the base.