Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion

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

(i) Additivity on $\mathcal{M}^{+}$. Let $f,g\in\mathcal{M}^{+}$. Then $f+g\in\mathcal{M}^{+}$ and

$$\int_{X}(f+g)\,d\mu=\int_{X}f\,d\mu+\int_{X}g\,d\mu .$$

(ii) Series in $\mathcal{M}^{+}$. Let $(f_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathcal{M}^{+}$, and let $\sum_{k}f_{k}:X\to[0,\infty]$ be the function with

$$\Bigl(\sum_{k}f_{k}\Bigr)(x)=\sum_{k\in\mathbb{N}}f_{k}(x)\qquad(x\in X),$$

the sum in $[0,\infty]$ of the sequence $(f_{k}(x))_{k}$. Then $\sum_{k}f_{k}\in\mathcal{M}^{+}$ and

$$\int_{X}\sum_{k}f_{k}\,d\mu=\sum_{k\in\mathbb{N}}\int_{X}f_{k}\,d\mu .$$

(iii) Positive and negative parts. Let $f:X\to\mathbb{R}$ be a function. Then

$$f=f^{+}-f^{-},\qquad |f|=f^{+}+f^{-}.$$

Moreover, for $x\in X$: if $f(x)\ge0$, then $f^{+}(x)=f(x)$ and $f^{-}(x)=0$; if $f(x)\le0$, then $f^{+}(x)=0$ and $f^{-}(x)=-f(x)$.

(iv) Integrability criterion. Let $f:X\to\mathbb{R}$ be measurable. Then $|f|,f^{+},f^{-}\in\mathcal{M}^{+}$, with, for their integrals as elements of $\mathcal{M}^{+}$,

$$\int_{X}|f|\,d\mu=\int_{X}f^{+}\,d\mu+\int_{X}f^{-}\,d\mu ,$$

and

$$f\in\mathcal{L}^{1}(\mu)\iff\int_{X}|f|\,d\mu<\infty,$$

the integral of $|f|$ as an element of $\mathcal{M}^{+}$.

(v) Domination. Let $f:X\to\mathbb{R}$ be measurable and $g\in\mathcal{L}^{1}(\mu)$ with $|f(x)|\le g(x)$ for every $x\in X$. Then $f\in\mathcal{L}^{1}(\mu)$.

(vi) Nonnegative real-valued functions. Let $f:X\to\mathbb{R}$ be measurable with $f(x)\ge0$ for every $x\in X$. Then $f\in\mathcal{M}^{+}$, and $f\in\mathcal{L}^{1}(\mu)$ if and only if $\int_{X}f\,d\mu<\infty$ for the integral of $f$ as an element of $\mathcal{M}^{+}$; in that case this integral is a real number (Small Facts about [0,∞]: Zero, Finiteness and Sums of Zeros (ii)), and it equals the integral of $f$ as an element of $\mathcal{L}^{1}(\mu)$.

(vii) Linearity. Let $f,g\in\mathcal{L}^{1}(\mu)$ and $a,b\in\mathbb{R}$. Then $af+bg\in\mathcal{L}^{1}(\mu)$ and

$$\int_{X}(af+bg)\,d\mu=a\int_{X}f\,d\mu+b\int_{X}g\,d\mu .$$

(viii) Finite linear combinations. Let $N\in\mathbb{N}$, let $[N]$ be the segment $\{1,\dots,N\}$, let $c:[N]\to\mathbb{R}$, $j\mapsto c_{j}$, be a function, and let $\Phi:[N]\times X\to\mathbb{R}$ be a function such that, for every $j\in[N]$, the function $f_{j}=\Phi(j,\cdot)$, $x\mapsto\Phi(j,x)$, lies in $\mathcal{L}^{1}(\mu)$. Let $\sum_{j=1}^{N}c_{j}f_{j}:X\to\mathbb{R}$ be the function with

$$\Bigl(\sum_{j=1}^{N}c_{j}f_{j}\Bigr)(x)=\sum_{j=1}^{N}c_{j}\Phi(j,x)\qquad(x\in X),$$

the finite sum in $\mathbb{R}$. Then $\sum_{j=1}^{N}c_{j}f_{j}\in\mathcal{L}^{1}(\mu)$ and

$$\int_{X}\sum_{j=1}^{N}c_{j}f_{j}\,d\mu=\sum_{j=1}^{N}c_{j}\int_{X}f_{j}\,d\mu .$$

(ix) Monotonicity. Let $f,g\in\mathcal{L}^{1}(\mu)$ with $f(x)\le g(x)$ for every $x\in X$. Then

$$\int_{X}f\,d\mu\le\int_{X}g\,d\mu .$$

(x) Absolute value. Let $f\in\mathcal{L}^{1}(\mu)$. Then $|f|\in\mathcal{L}^{1}(\mu)$ and

$$\Bigl|\int_{X}f\,d\mu\Bigr|\le\int_{X}|f|\,d\mu,$$

the integral of $|f|$ as an element of $\mathcal{L}^{1}(\mu)$.