I am currently studying Rudin's RCA book and I have a question about Theorem 2.20, where the author constructs Lebesgue measure on $\mathbb{R}^k$.
Here are the definitions and the notations I am basing my question on :
- $C_c(\mathbb{R}^k)$ is the set of complex, continuous functions on $\mathbb{R}^k$ with compact support,
- A $k$-cell is any Cartesian product of intervals (all open, closed or semi-open),
- The $\delta$-box with corner at $a$ refers to the set $Q(a,\delta) = \{x \in \mathbb{R}^k : \quad a_i \le x_i < a_i + \delta, \quad i=1,\cdots,k \}$,
- $P_n$ is the set of points with integral multiples of $2^{-n}$ as coordinates,
- $\Omega_n$ is the set of all $2^{-n}$-boxes with corners in $P_n$
I have a problem with the first part of the proof which states that, if we take a real function $f \in C_c(\mathbb{R}^k)$ and $W$ an open k-cell that contains the support of $f$, and we let $\varepsilon> 0$, then there exist an integer $N\ge 1$ and functions $g, h:\mathbb{R}^k\rightarrow\mathbb{R}$ with supports in $W$ such that :
- $g$ and $h$ are constants on each box in $\Omega_N$,
- $g \le f \le h$,
- $h-g < \varepsilon$
Rudin justifies it with the uniform continuity of $f$ and moves on, but I can't come up with an explicit construction that satisfies all the properties. I checked the accepted comment here and my attempt is as follows.
The definition of uniform continuity is$$\forall \eta >0 \quad \exists \delta > 0 \quad \forall x,y \in \bar{W} \quad \|x-y\|<\delta \implies |f(x)-f(y)|\le\eta$$
We take $\eta = \frac{\varepsilon}{3}$. There exists an integer $N \ge 1 $ such that $\sqrt k2^{-N} < \delta$ (the factor $\sqrt{k}$ shows up because I use the Euclidean norm).
Now, let $K$ be the compact support of $f$, if $Q=Q(a, 2^{-N}) \in \Omega_N$ and $Q\subset K$, i take, for every $x \in Q$,$g(x) = f(a) -\frac{\varepsilon}{3}$ and $h(x) = f(a) + \frac{\varepsilon}{3}$. If $Q$ is not a subset of $K$, there are two cases. The first case is when its intersection with $K$ is empty and I can set $g$ and $h$ equal to zero on $Q$. The second case is when the intersection is not empty and I don't know how to define $g$ and $h$ without having their support (eventually) stepping outside $W$.
Thank you for reading.