Approximation of Measurable Functions by Dyadic Simple Functions

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F})$ be a measurable space, and let $\mathcal{S}=\mathcal{S}(X,\mathcal{F})$ and $\mathcal{S}^{+}=\mathcal{S}^{+}(X,\mathcal{F})$ be the set of simple functions of Simple Functions, Nonnegative Simple Functions, and the Simple Integral (ii) and the set of nonnegative simple functions of Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iii). For $y\in\mathbb{R}$, $|y|$ is its absolute value; for $y,z\in\mathbb{R}$, $\max\{y,z\}$ is formed in $\mathbb{R}$ as in The Maximum and Minimum of Two Elements of a Total Order (i); for $m\in\mathbb{N}$, $2^{m}$ is the power of $2$ in $\mathbb{R}$ and $2^{-m}=1/2^{m}$. A map $X\to\mathbb{R}$ with values $\ge0$ is also a map $X\to[0,\infty]$, by The Extended Real Line, Its Order, and [0,∞] (ii). For a map $g:X\to[0,\infty]$ and $m\in\mathbb{N}$, $s_{m}(g):X\to\mathbb{R}$ is the dyadic approximation of $g$ of order $m$.

(i) Let $f:X\to[0,\infty]$ be measurable as a map into $[0,\infty]$. Then $s_{m}(f)\in\mathcal{S}^{+}$ for every $m\in\mathbb{N}$, and for every $x\in X$,

$$s_{m}(f)(x)\le s_{m+1}(f)(x)\le f(x)\quad\text{for every }m\in\mathbb{N},\qquad f(x)=\sup\{s_{m}(f)(x):m\in\mathbb{N}\},$$

where $\le$ is the order of The Extended Real Line, Its Order, and [0,∞] (iii), which between real numbers is the order of $\mathbb{R}$, and the supremum is that of the subset $\{s_{m}(f)(x):m\in\mathbb{N}\}$ of $[0,\infty]$, nonempty as $1\in\mathbb{N}$ by Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication (ii), which exists by Order and Suprema in the Extended Real Line (iii) and lies in $[0,\infty]$ by Order and Suprema in the Extended Real Line (ix).

(ii) Let $K\in\mathbb{R}$ and let $f:X\to\mathbb{R}$ be measurable as a map into $\mathbb{R}$ with $0\le f(x)\le K$ for every $x\in X$, so that $f$ is also a map into $[0,\infty]$, by The Extended Real Line, Its Order, and [0,∞] (ii). Then $s_{m}(f)\in\mathcal{S}^{+}$ for every $m\in\mathbb{N}$, and $0\le f(x)-s_{m}(f)(x)<2^{-m}$ for every $x\in X$ and every $m\in\mathbb{N}$ with $m\ge K$.

(iii) Let $f:X\to\mathbb{R}$ be measurable as a map into $\mathbb{R}$, let $f^{+},f^{-}:X\to\mathbb{R}$ be the maps with $f^{+}(x)=\max\{f(x),0\}$ and $f^{-}(x)=\max\{-f(x),0\}$, and for $m\in\mathbb{N}$ let $t_{m}:X\to\mathbb{R}$ be the map with $t_{m}(x)=s_{m}(f^{+})(x)-s_{m}(f^{-})(x)$. Then $t_{m}\in\mathcal{S}$ and $|t_{m}(x)|\le|f(x)|$ for every $m\in\mathbb{N}$ and every $x\in X$, and for every $x\in X$ the sequence $(t_{m}(x))_{m\in\mathbb{N}}$ converges to $f(x)$.