# Doubt in understanding theorem 4.1 Real analysis Stein and Shakarachi

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.

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

How can this partition be visualised?

Any help will be appreciated.

• Please post self-contained questions. Images aren't a shortcut to typing math. Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be searched and are not view-able to some, such as those who use screen readers. – GNUSupporter 8964民主女神 地下教會 Feb 4 at 13:02
• Ok Sir I will Write. Sorry I do not know earlier this. – SRJ Feb 4 at 13:03
• Please ping me again when you've finished typing it. I will replace my downvote with an upvote. Btw, I seize this opportunity to introduce you to Approach 0, Math.SE's search engine by LaTeX. Once you get used to its convenience, you'll understand my anti pic-question stance. – GNUSupporter 8964民主女神 地下教會 Feb 4 at 13:07

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

• $$Q_N$$ 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 $$F_{N,M}(x) := \sum_{\ell=0}^{NM} \frac{\ell}{M} \chi_{E_{\ell,M}}(x).$$

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.)