Finite Signed Measures from Finite Measures and from Integrals

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F})$ be a measurable space, let $\mathcal{G}\subseteq\mathcal{F}$ be a $\sigma$-algebra on $X$, and for a map $\rho$ with domain $\mathcal{F}$ let $\rho|_{\mathcal{G}}$ be its restriction to $\mathcal{G}$.

(i) Differences of finite measures. Let $\mu_{1}$ and $\mu_{2}$ be finite measures on $(X,\mathcal{F})$. Then the map $\mu_{1}-\mu_{2}$ on $\mathcal{F}$, $A\mapsto\mu_{1}(A)-\mu_{2}(A)$, is a finite signed measure on $(X,\mathcal{F})$.

(ii) Finite measures. Every finite measure on $(X,\mathcal{F})$ is a finite signed measure on $(X,\mathcal{F})$.

(iii) Restriction to a sub-$\sigma$-algebra. Let $\mu$ be a measure on $(X,\mathcal{F})$. Then $\mu|_{\mathcal{G}}$ is a measure on $(X,\mathcal{G})$; it is finite if $\mu$ is finite, and a probability measure if $\mu$ is one. If $\nu$ is a finite signed measure on $(X,\mathcal{F})$, then $\nu|_{\mathcal{G}}$ is a finite signed measure on $(X,\mathcal{G})$.

(iv) Indefinite integrals. Let $\mu$ be a measure on $(X,\mathcal{F})$ and $f\in\mathcal{L}^{1}(\mu)$. Then the map $\nu_{f}$ on $\mathcal{G}$,

$$\nu_{f}(A)=\int_{A}f\,d\mu ,$$

is a finite signed measure on $(X,\mathcal{G})$ that is absolutely continuous with respect to the measure $\mu|_{\mathcal{G}}$ ((iii)). If moreover $f(x)\ge0$ for every $x\in X$, then $\nu_{f}$ is a finite measure on $(X,\mathcal{G})$ that is absolutely continuous with respect to $\mu|_{\mathcal{G}}$.