A bounded, continuous function on $\mathbb{R}^d$ is the pointwise limit of a bounded sequence of linear combinations of Borel indicators

Question

Where can I find a proof of the following fact?

Any bounded, continuous real-valued function on $\mathbb{R}^d$ is the pointwise limit of a bounded sequence of linear combinations of indicators of Borel sets

Answer 1

A stronger result is true. You can get uniform limit instead of pointwise limit. Just define $f_n(x)=\sum_j \frac {j-1} n I_{f^{-1}(\frac {j-1} n,\frac j n)}$. Since $f$ is bounded the sum is actually a finite sum. We have $|f_n(x)-f(x)| \leq \frac 1 n$ for all $x$. [Since $f$ is continuous the inverse image of any Borel set is a Borel set].

Answer 2

I would suggest looking in Rudin's Real & Complex Analysis or a similar style book (Royden, Folland, etc. come to mind). I know that there are variants of this in Chapter 1 and Chapter 2 of Rudin.

I'm not sure that this is really a named theorem, since it's fairly elementary. One quick way to prove it is to approximate

$$f \approx \sum_{Q \in \mathcal{D}} \chi_Q f_Q$$

where $\mathcal{D}$ is a collection of disjoint sets $Q$ which cover $\mathbb{R}^d$. (For example, $\mathcal{D}$ is the standard dyadic grid). Then if we consider a sequence of such collections where the maximum diameter goes to zero, it's easy to show that the approximants are bounded and converge to $f$ pointwise.