Doubt in understanding theorem 4.1 Real analysis Stein and Shakarachi

measurable-functionsmeasure-theoryproof-explanationreal-analysisvisualization

I came across the following theorem. I understood that $F_k(x)\to f(x)$ pointwise as $k\to \infty$, but I do not understand how $E_{l,j} $ is defined over a range of $F_k$. I am not able to visualise.

Please suggest me some visualisation.

Edit:

Theorem 4.1 Suppose $f$ is a non-negative measurable function on $\Bbb{R}^d$.
Then there exists an increasing sequence of non-negative simple functions
$(\varphi_k)_k$
that converges pointwise to $f$, namely,
$$\varphi_k(x)\leq \varphi_{k+1}
\text{ and } \lim_{k\to \infty}=f(x) ,\forall x$$

I understand

$$F_k(x)=
\begin{cases}
f(x) &\text{ if } x\in Q_k f(x)\leq k \\
k &\text{ if } x\in Q_k f(x)> k \\
0 &\text{ otherwise},
\end{cases}
$$
where $Q_k$ denote cube centered ar origin and of side length $k$.

I do not understand following

We partition the range of $F_k$ namely $[0,k]$ as follows. For fixed $k,j>1$,
$$E_{l,j}=\left\{x\in Q_k \biggm| \frac{l}{j} < F_k(x)\leq \frac{l+1}{j}\right\} \quad \forall 0\leq l<kj.$$

How can this partition be visualised?

theorem photo

enter image description here

Any help will be appreciated.

Best Answer

The author is approximating $f$ from below by step function $F_{N,M}$ of step height $1/M$ defined on $[-N,N]$. Intuitively, as $N,M \to \infty$ (wider domain and finer step size), we should expect step function approximation $F_{N,M}$ converge pointwisely to $f$ everywhere.

In $$E_{\ell,M} = \{x\in Q_N \mid \ell/M<F_N(x)\leq (\ell+1)/M \}, \qquad 0\leq \ell< NM,$$

  • $Q_N$ truncates the domain to $[-N,N]$;
  • $F_N$ truncates the range to $[0,N]$;
  • $M$ controls the step height, so that $[0,N]$ is divided evenly into $NM$ sufficiently small half-open half-closed intervals (except at $y=0$).
  • $\ell$ iterates the through these subintervals.
  • The preimage of each subinterval is denoted by $E_{\ell,M}$.
  • Note that for each $M \in \Bbb{N}^*$, $\{E_{\ell,M}\}_\ell$ is a disjoint collection subsets, and that $F[E_{\ell,M}]$ can be approximated, by the construction of $E_{\ell,M}$, by $(\ell/M,(\ell+1)/M]$.

Now, we have the lower step function approximation $$F_{N,M}(x) := \sum_{\ell=0}^{NM} \frac{\ell}{M} \chi_{E_{\ell,M}}(x).$$

step function lower approximation
Click to view live demo.

Exercise: Find a measurable function $f$ defined on $\Bbb{R}$ so that $\bigcup_{N=0}^\infty\bigcup_{M=0}^\infty\bigcup_{l=0}^{NM} E_{\ell,M} \subsetneq \Bbb{R}$? (In words, think about the case when the above construction of $\chi_{E_{\ell,M}}$ fails to cover the domain of $f$ no matter how "good" our approximation $F_{N,M}$ is, i.e. how large $N,M$ are.)

Related Question