Increasing sequence of simple functions to a bounded measurable function f

Assume $|f(x)|\leq M$ , where $M\in\mathbb{N}$ . Consider $\displaystyle g_k=\sum_{i=-M2^k+1}^{M2^k}\frac{i-1}{2^k}\chi_{\{\frac{i-1}{2^k}<f\leq\frac{i}{2^k}\}}.$
Thus $\displaystyle g_{k+1}=\sum_{i=-M2^{k+1}+1}^{M2^{k+1}}\frac{i-1}{2^{k+1}}\chi_{\{\frac{i-1}{2^{k+1}}<f\leq\frac{i}{2^{k+1}}\}}.$

For $x\in\{\frac{i-1}{2^k}<f\leq\frac{i}{2^k}\}$ , $g_k(x)=\frac{i-1}{2^k}$ .

The above set is equal to $\{\frac{2i-2}{2^{k+1}}<f\leq\frac{2i}{2^{k+1}}\}$ , so $g_{k+1}(x)\geq\frac{2i-2}{2^{k+1}}=g_k(x)$ .

$|g_k(x)-f(x)|\leq\frac{1}{2^k}\to 0$ as $k\to\infty$ . Hence $g_k$ converges to $f$ everywhere.

Author: mathtuition88

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