Integrals with Respect to the Restriction of a Measure to a Sub-σ-Algebra

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space, let $\mathcal{G}\subseteq\mathcal{F}$ be a $\sigma$-algebra on $X$, and let $\mu|_{\mathcal{G}}$ be the restriction of $\mu$ to $\mathcal{G}$, a measure on $(X,\mathcal{G})$ (Finite Signed Measures from Finite Measures and from Integrals (iii)).

(i) Nonnegative functions. Let $f\in\mathcal{M}^{+}(X,\mathcal{G})$. Then $f\in\mathcal{M}^{+}(X,\mathcal{F})$ and

$$\int_{X}f\,d(\mu|_{\mathcal{G}})=\int_{X}f\,d\mu .$$

(ii) Real functions. Let $f:X\to\mathbb{R}$ be $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable. Then $f$ is $\mathcal{F}/\mathcal{B}(\mathbb{R})$-measurable, $f\in\mathcal{L}^{1}(X,\mathcal{G},\mu|_{\mathcal{G}})$ if and only if $f\in\mathcal{L}^{1}(\mu)$, and in that case

$$\int_{X}f\,d(\mu|_{\mathcal{G}})=\int_{X}f\,d\mu .$$

(iii) Nonnegative functions over sets of $\mathcal{G}$. Let $G\in\mathcal{G}$ and $f\in\mathcal{M}^{+}(X,\mathcal{G})$. Then

$$\int_{G}f\,d(\mu|_{\mathcal{G}})=\int_{G}f\,d\mu .$$

(iv) Real functions over sets of $\mathcal{G}$. Let $G\in\mathcal{G}$ and let $f:X\to\mathbb{R}$ be $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable. Then $f$ is integrable on $G$ with respect to $\mu|_{\mathcal{G}}$ if and only if it is integrable on $G$ with respect to $\mu$, and then $\int_{G}f\,d(\mu|_{\mathcal{G}})=\int_{G}f\,d\mu$.