Integrals of Functions over a Measurable Subset

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $A\in\mathcal{F}$. Let

$$\mathcal{F}_{A}=\{B\in\mathcal{F}:B\subseteq A\},\qquad \mu_{A}=\mu|_{\mathcal{F}_{A}},$$

so that $\mathcal{F}_{A}$ is a $\sigma$-algebra on $A$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (xi)) and $\mu_{A}$, the restriction of $\mu$ to $\mathcal{F}_{A}$, is a function $\mathcal{F}_{A}\to[0,\infty]$ with $\mu_{A}(B)=\mu(B)$ for $B\in\mathcal{F}_{A}$ (Restricting a Measure to a Measurable Set). For a function $f$ on $X$, $f|_{A}$ is its restriction to $A$, the function on $A$ with $f|_{A}(x)=f(x)$.

(i) The restricted measure space. The restriction of $(X,\mathcal{F},\mu)$ to $A$ is the measure space $(A,\mathcal{F}_{A},\mu_{A})$ ($\mu_{A}$ is a measure by Restricting a Measure to a Measurable Set (i)).

(ii) Nonnegative functions. For $f\in\mathcal{M}^{+}(X,\mathcal{F})$, the set of measurable functions into $[0,\infty]$, the integral of $f$ over $A$ with respect to $\mu$ is the integral

$$\int_{A}f\,d\mu=\int f|_{A}\,d\mu_{A}\in[0,\infty]$$

(here $f|_{A}\in\mathcal{M}^{+}(A,\mathcal{F}_{A})$ by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (viii)).

(iii) Integrability on $A$. A function $f\in\mathrm{Meas}(X,\mathcal{F};\mathbb{R},\mathcal{B}(\mathbb{R}))$, the set of measurable functions $X\to\mathbb{R}$ of Measurable Map, and the Set of Measurable Maps (ii), is integrable on $A$ with respect to $\mu$ if

$$f|_{A}\in\mathcal{L}^{1}(A,\mathcal{F}_{A},\mu_{A}),$$

the set of integrable functions for $(A,\mathcal{F}_{A},\mu_{A})$ (here $f|_{A}$ is measurable by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (viii)).

(iv) Integrals of integrable functions. If $f\in\mathrm{Meas}(X,\mathcal{F};\mathbb{R},\mathcal{B}(\mathbb{R}))$ is integrable on $A$, the integral of $f$ over $A$ with respect to $\mu$ is the real number

$$\int_{A}f\,d\mu=\int f|_{A}\,d\mu_{A},$$

the integral of $f|_{A}\in\mathcal{L}^{1}(A,\mathcal{F}_{A},\mu_{A})$.