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

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.