2
$\begingroup$

Let $ f: \mathbb R \to \mathbb R $ a non-negative bounded measurable function. Prove that there exists a sequence of simple non-negative functions $ (f_n)_{n \in \mathbb N} $ such that $ f_n \to f$ uniform.

Searching on Wikipedia I found the following http://en.wikipedia.org/wiki/Simple_function

but I can't understand why the converge is uniform.

Any help?

  • 0
    The key point behind the uniform part of this statement lies in the boundedness of $f$.2012-06-28

2 Answers 2

4

Hint: if $A \le f \le B$, split up $[A,B]$ into $n$ equal intervals and choose a value for $f_n$ in each one.

0

I think the way it works is like this:

  • $f$ is measurable $\Rightarrow$ a series of simple functions $\{f_n\}$ exists that converges uniformly to $f$.
  • a series of simple function $\{f_n\}$ exists that converges uniformly to $f$ $\Rightarrow$ $f$ is measurable .
  • a series of simple function $\{g_n\}$ exists that converges non-uniformly to $f$ $\Rightarrow$ $f$ is measurable $\Rightarrow$ a series of simple functions $\{f_n\}$ exists (different from $g_n$) that converges uniformly to $f$.