Integrals over Measurable Subsets

In the setting of Measurable Functions and the Lebesgue Integral: Notation, let $(X,\mathcal{F},\mu)$ be a measure space.

(i) Integrals over a subset as integrals over $X$: nonnegative functions. Let $f\in\mathcal{M}^{+}$ and $A\in\mathcal{F}$. Then $f\mathbf{1}_{A}\in\mathcal{M}^{+}$ and, for the integral over $A$ of $f\in\mathcal{M}^{+}$,

$$\int_{A}f\,d\mu=\int_{X}f\mathbf{1}_{A}\,d\mu .$$

(ii) Integrals over a subset as integrals over $X$: integrable functions. Let $f:X\to\mathbb{R}$ be measurable, let $A\in\mathcal{F}$, and let $f\mathbf{1}_{A}$ be the product of the real-valued functions $f$ and $\mathbf{1}_{A}$. Then $f$ is integrable on $A$ if and only if $f\mathbf{1}_{A}\in\mathcal{L}^{1}(\mu)$, and in that case, for the integral over $A$ of an integrable function and the integral of $f\mathbf{1}_{A}\in\mathcal{L}^{1}(\mu)$,

$$\int_{A}f\,d\mu=\int_{X}f\mathbf{1}_{A}\,d\mu .$$

(iii) Nonnegative real-valued functions. Let $f:X\to\mathbb{R}$ be measurable with $f(x)\ge0$ for every $x\in X$, and let $A\in\mathcal{F}$. Then $f\in\mathcal{M}^{+}$ (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (vii)), and $f$ is integrable on $A$ if and only if $\int_{A}f\,d\mu<\infty$ for the integral over $A$ of $f\in\mathcal{M}^{+}$; in that case this integral is a real number, and it equals the integral of $f$ over $A$ as an integrable function.

(iv) Zero extensions. Let $A\in\mathcal{F}$, and let $g:A\to\mathbb{R}$ and $\tilde g:X\to\mathbb{R}$ be functions with

$$\tilde g(x)=g(x)\quad(x\in A),\qquad\tilde g(x)=0\quad(x\in X\setminus A).$$

Then $g$ is measurable for $\mathcal{F}_{A}$ if and only if $\tilde g$ is measurable for $\mathcal{F}$; in that case $g\in\mathcal{L}^{1}(A,\mathcal{F}_{A},\mu_{A})$ if and only if $\tilde g\in\mathcal{L}^{1}(\mu)$, and then, for the integrals of these elements of $\mathcal{L}^{1}(A,\mathcal{F}_{A},\mu_{A})$ and $\mathcal{L}^{1}(\mu)$,

$$\int g\,d\mu_{A}=\int_{X}\tilde g\,d\mu .$$

Likewise, let $g:A\to[0,\infty]$ and $\tilde g:X\to[0,\infty]$ be functions with the same two properties. Then $g$ is measurable for $\mathcal{F}_{A}$ if and only if $\tilde g$ is measurable for $\mathcal{F}$, and in that case $\int g\,d\mu_{A}=\int_{X}\tilde g\,d\mu$, for the integrals of $g\in\mathcal{M}^{+}(A,\mathcal{F}_{A})$ and $\tilde g\in\mathcal{M}^{+}$.

(v) The whole space. For $f\in\mathcal{M}^{+}$, the integral of $f$ over $X$ of Integrals of Functions over a Measurable Subset (ii) is the integral $\int_{X}f\,d\mu$ of $f\in\mathcal{M}^{+}$. A measurable function $f:X\to\mathbb{R}$ is integrable on $X$ if and only if $f\in\mathcal{L}^{1}(\mu)$, and in that case the integral of $f$ over $X$ of Integrals of Functions over a Measurable Subset (iv) is the integral $\int_{X}f\,d\mu$ of $f\in\mathcal{L}^{1}(\mu)$.

(vi) Integrable functions are integrable on every measurable subset. Every $f\in\mathcal{L}^{1}(\mu)$ is integrable on every $A\in\mathcal{F}$.

(vii) Additivity over disjoint sets. Let $A,B\in\mathcal{F}$ with $A\cap B=\emptyset$. Then $A\cup B\in\mathcal{F}$, and for every $f\in\mathcal{M}^{+}$, with the integrals over subsets of $f\in\mathcal{M}^{+}$ and the sum of $[0,\infty]$, and for every $f\in\mathcal{L}^{1}(\mu)$, which is integrable on $A$, on $B$ and on $A\cup B$ by (vi), with the integrals over subsets of an integrable function and the sum of $\mathbb{R}$,

$$\int_{A\cup B}f\,d\mu=\int_{A}f\,d\mu+\int_{B}f\,d\mu .$$

(viii) Monotonicity in the set. Let $f\in\mathcal{M}^{+}$ and $A,B\in\mathcal{F}$ with $A\subseteq B$. Then, for the integrals over $A$ and $B$ of $f\in\mathcal{M}^{+}$,

$$\int_{A}f\,d\mu\le\int_{B}f\,d\mu .$$

(ix) Sets of measure zero. Let $A\in\mathcal{F}$ with $\mu(A)=0$. Then $\int_{A}f\,d\mu=0$ for every $f\in\mathcal{M}^{+}$, with its integral over $A$ as an element of $\mathcal{M}^{+}$; and every measurable function $f:X\to\mathbb{R}$ is integrable on $A$, with $\int_{A}f\,d\mu=0$ for its integral over $A$ as an integrable function.