How to show that a measurable function on $R^d$ can be approximated by step functions?

Question

In Stein's Book: Real Analysis, Theorem 4.3 says that any measurable function $f$ on $R^d$ can be approximated by step functions. But the proof provided there only show that when $f=\chi_E$, with $m(E)<+\infty$ can be approximated by step function $\psi_k$ (a.e.). How to complete the argument?

Answer 1

I would like to share my answer, it is a litter long, and I will just guide the proof:

step 1: For $f=\chi_E$, with $m(E)<\infty$, see Stein's Book Thm4.3: there exist step function $\phi_k\to\chi_E$ a.e. in $R^d$;

step 2:. For general simple function $\psi=\sum_{l=1}^n a_l\chi_{E_{l}}(x)$ (note that by definition $m(E_l)<\infty$, $\forall l$), there exists step function $\phi_{k}$, such that $\phi_{k}\to\psi$ a.e. in $R^d$;

step 3: By Stein's book Thm4.2, for measurable function $f$ on $R^d$, there exists simple function $\psi_k\to f$ point-wise in $R^d$. Thus, by step 2. there exist step function $ \phi_{k,j}\to\psi_k$, $\forall k$. For each $k$, we can select a cubic $Q_k$ centered at the origin with length of side k. We see that $\phi_{k,j}\chi_{Q_k}\to\psi_k$,$\forall k$, so we can replace $\phi_{k,j}$ by $\phi_{k,j}\chi_{Q_k}$.

step 4: Apply Egorov's Thm on $Q_k$ to select $J(k)$, such that $$|\psi_k-\phi_{k,J(k)}|<\frac{1}{k},\quad\forall x\in A_k.$$ where $A_k$ is contained in the union of $ Q_k \text{and }\cup_{l=1}^{\infty} E_l$ with $ m(Q_k\cup\cup_{l=1}^{\infty} E_l\backslash A_k)<\frac{1}{2^k}$.

step 5: Let $E=\limsup_{k=1}^\infty ((Q_k\cup\cup_{l=1}^{\infty} E_l\backslash A_k)$, prove that $m(E)=0$, and $\phi_{k,J(k)}\to f$ a.e. in $E^c$.

Answer 2

Here's another approach.

1. Let $E$ be a measurable set, and $\epsilon>0$. Then there exists a step function $\psi$ s.t. $m\left(\big\{\,x\,\big\vert\,\psi(x)\neq\chi_E(x)\big\}\right)\le\epsilon$.
2. Let $\phi$ be a simple function and $\epsilon>0$. Then there exists a step function $\psi$ s.t. $m\left(\big\{\,x\,\big\vert\,\psi(x)\neq\phi(x)\big\}\right)\le\epsilon$.
3. Let $(f_n)$ and $(g_n)$ be two sequences of functions such that $f_n\to f$ as $n\to\infty$, and $\sum_{n=1}^\infty m_*\big(\big\{\,x\,\big\vert\,g_n(x)\neq f_n(x)\,\}\big)<+\infty$. Then $g_n\to g$ a.e. (Hint: it's easier if you apply the Borel–Cantelli lemma in Stein, RA, Chapter 1, Exercise 16)
4. Note that $f$ could be pointwise approximated by simple functions, then apply 2,3.