Definitions: An integrable simple function $f = a_1 \chi_{A_1} + \ldots + a_n \chi_{A_n},$ where $\chi_{A_k}$ is an indicator function and the $A_j$'s are disjoint measurable sets of finite measure. A step function is a piecewise constant function with a finite number of pieces (basically it is an ISF with intervals as the sets).
Prove that for any ISF $f$ and $\epsilon > 0,$ there is a step function $g$ such that $\displaystyle\int |f-g| < \epsilon.$
The question says to use the approximation property for measurable sets (measurable sets can be approximated by open and closed sets) and the fact that $m(A\triangle B) = \displaystyle\int |\chi_A – \chi_B|,$ where $A\triangle B$ is the symmetric difference.
Basically, I'm having trouble finding a finite set of intervals for this approximation – any advice would be appreciated.
Best Answer
I am assuming that your domain is $\mathbb{R}$ with the Lebesgue measure.
As is typical with these sort of things, first we try to approximate a single indicator function $1_A$.
Let $\epsilon>0$. If $A$ has finite measure, then there exists an open $U$ such that $m (U \setminus A) < \frac{1}{2} \epsilon$.
Any open $U \subset \mathbb{R}$ can be written as $U = \cup_{k=1}^\infty I_k$, where $\{I_k\}$ is a countable collection of disjoint open intervals. Since the $I_k$ are disjoint we have $m U = \sum_{k=1}^\infty m I_k$. Choose a finite collection $\{I_k\}_{k \in J}$ such that $mU -\sum_{k \in J} m I_k = m ( U\setminus \cup_{k \in J} I_k) < \frac{1}{2} \epsilon$.
Since $\cup_{k \in J} I_k \subset U$ and $A \subset U$, it follows that $\cup_{k \in J} I_k \setminus A \subset U \setminus A$, and $A \setminus \cup_{k \in J} I_k \subset U \setminus \cup_{k \in J} I_k$, hence $A \triangle \cup_{k \in J} I_k \subset (U \setminus A )\cup (U \setminus \cup_{k \in J} I_k)$, from which it follows that $m(A \triangle \cup_{k \in J} I_k ) < \epsilon$. Hence $\int |1_A -1_{\cup_{k \in J} I_k}| = m(A \triangle \cup_{k \in J} I_k ) < \epsilon$. Hence the characteristic function of any $A$ with finite measure can be approximated by the characteristic function of a finite collection of disjoint intervals.
I need some notation: Let $B$ ($=\cup_{k \in J} I_k$ above) be the finite union of intervals that approximates $A$ as above.
Suppose we are given $f=\sum_k a_k 1_{A_k}$, and choose $\epsilon>0$. Now choose the unions of intervals $B_k$ such that $m(A_k \triangle B_k) < \frac{1}{\sum_k |a_k|} \epsilon$. Then $$\int |f-\sum_k a_k 1_{B_k}| = \int |\sum_k a_k (1_{A_k}-1_{B_k})| \leq \sum_k |a_k| \int|1_{A_k}-1_{B_k}| < \epsilon$$ Since $\sum_k a_k 1_{B_k}$ is a step function, we are finished.