1
$\begingroup$

On the Real Analysis - Modern Techniques and Their Application (second edition) by Gerald Folland, page 47 i found this theorem: "Let $f$ a measurable function. Then exists a sequence $(\phi_n)_{n \in \mathbb{N}}$ of simple functions such that: (a) $\phi_n \to f$ pointwise; (b) $\phi_n \to f$ uniformly on any set on which $f$ is bounded".

I need a proof step by step of this theorem. Can someone suggest me a more didactic book?

  • 1
    _Real Analysis_ by N.L. Carothers provides a proof of (a), but leaves a few steps to the reader. _Measure and Integral_ by Zygmund and Wheeden is more explicit, I think, but also more formal. Neither book provides a proof that $\phi_n \rightarrow f$ uniformly on a set on which $f$ is bounded, but it's pretty clear from the construction of $\phi_n$. Think about this: for any $n$, what is $\sup_x |f(x) - \phi_n(x)|$ if $f$ is bounded?2011-11-03

1 Answers 1