The Lebesgue Integral and Null Sets

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

(i) Integrals over null sets. Let $f\in\mathcal{M}^{+}$ and let $N\subseteq X$ be a $\mu$-null set with $f(x)=0$ for every $x\in X\setminus N$. Then

$$\int_{X}f\,d\mu=0 .$$

(ii) Almost-everywhere comparison. Let $u,v\in\mathcal{M}^{+}$, with their integrals as elements of $\mathcal{M}^{+}$. If $u(x)\le v(x)$ for $\mu$-almost every $x$, then

$$\int_{X}u\,d\mu\le\int_{X}v\,d\mu ;$$

if $u(x)=v(x)$ for $\mu$-almost every $x$, then $\int_{X}u\,d\mu=\int_{X}v\,d\mu$.

(iii) Functions equal almost everywhere to an integrable function. Let $f\in\mathcal{L}^{1}(\mu)$, and let $g:X\to\mathbb{R}$ be measurable with $f(x)=g(x)$ for $\mu$-almost every $x$. Then $g\in\mathcal{L}^{1}(\mu)$ and, for the integrals of $f$ and $g$ as elements of $\mathcal{L}^{1}(\mu)$,

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

(iv) Vanishing integrals. Let $f\in\mathcal{M}^{+}$. Then $\{x\in X:f(x)>0\}\in\mathcal{F}$, and

$$\int_{X}f\,d\mu=0\iff\mu\bigl(\{x\in X:f(x)>0\}\bigr)=0\iff f(x)=0\text{ for $\mu$-almost every }x .$$

(v) Finiteness almost everywhere. Let $f\in\mathcal{M}^{+}$ with $\int_{X}f\,d\mu<\infty$. Then $\{x\in X:f(x)=\infty\}\in\mathcal{F}$,

$$\mu\bigl(\{x\in X:f(x)=\infty\}\bigr)=0,$$

and $f(x)<\infty$ for $\mu$-almost every $x$.

(vi) Markov's inequality. Let $f\in\mathcal{M}^{+}$ and let $t>0$ be a real number. Then $A_{t}=\{x\in X:t\le f(x)\}$ belongs to $\mathcal{F}$ and, with the product of $[0,\infty]$,

$$t\,\mu(A_{t})\le\int_{X}f\,d\mu .$$

In the remaining clauses, let $(f_{m})_{m\in\mathbb{N}}$ be a sequence of measurable functions $f_{m}:X\to\mathbb{R}$, let $f:X\to\mathbb{R}$ be measurable and let $g\in\mathcal{L}^{1}(\mu)$; suppose that $f_{m}(x)\to f(x)$ for $\mu$-almost every $x$, and that for every $m\in\mathbb{N}$, $|f_{m}(x)|\le g(x)$ for $\mu$-almost every $x$.

(vii) Integrability. $f\in\mathcal{L}^{1}(\mu)$, and $f_{m}\in\mathcal{L}^{1}(\mu)$ for every $m\in\mathbb{N}$.

(viii) Convergence in mean. $|f_{m}-f|\in\mathcal{L}^{1}(\mu)$ for every $m\in\mathbb{N}$, and, for the integrals as elements of $\mathcal{L}^{1}(\mu)$,

$$\lim_{m\to\infty}\int_{X}|f_{m}-f|\,d\mu=0,$$

that is, the sequence $\bigl(\int_{X}|f_{m}-f|\,d\mu\bigr)_{m\in\mathbb{N}}$ converges to $0$.

(ix) Convergence of the integrals. For the integrals as elements of $\mathcal{L}^{1}(\mu)$,

$$\lim_{m\to\infty}\int_{X}f_{m}\,d\mu=\int_{X}f\,d\mu,$$

that is, the sequence $\bigl(\int_{X}f_{m}\,d\mu\bigr)_{m\in\mathbb{N}}$ converges to $\int_{X}f\,d\mu$.