Measure Theory/Approximations of Measurable Sets

Approximations of Measurable Sets

In this lesson we will see that every measurable set is "nearly" an open set, and also "nearly" a closed set.

Let $E\in {\mathcal {M}}$ be any measurable set, and let $\varepsilon \in {\mathbb {R}}^{+}$.

We will show that there exists an open set G such that $E\subseteq G$ and

$\lambda (G\smallsetminus E)<\varepsilon$

Exercise 1. Finite Measure Sets Approximated by Open Sets

Suppose that $\lambda (E)<\infty$ and $\varepsilon \in {\mathbb {R}}^{+}$. Prove that there exists an open set G such that $E\subseteq G$ and $\lambda (G\smallsetminus E)<\varepsilon$.

Hint: Use the $\varepsilon /2^{n}$ trick, and the fact that arbitrary unions of open sets are open sets.

Exercise 2. Infinite Measure Sets Approximated by Open Sets

Suppose that $\lambda (E)=\infty$ and $\varepsilon \in {\mathbb {R}}^{+}$. Again find an open set G as before.

Hint: The strategy is to take E and "do something to it" to get a finite-measure set. Apply the result for finite measures. Do this in a sequence which culminates in the desired set G.

Exercise 3. Approximation by Closed Sets

Infer from the previous exercises that, for $E\in {\mathcal {M}}$ and $\varepsilon \in {\mathbb {R}}^{+}$, there is a closed $F\subseteq E$ such that $\lambda (E\smallsetminus F)<\varepsilon$.

Hint: Apply the previous result to $E^{c}$.

Optional Exercise 4. Approximated by Open Sets Are Measurable

In fact it turns out that the converse is also true: If any set is approximated by open sets, then it must be measurable. Feel free to prove this if you would like a challenge problem -- however, we will not so often have use for this theorem. This exercise is therefore "merely" an exercise, for this course.

Exercise 5. Open Sets Are Countable Intervals

Prove that every open set is a countable union of open intervals.

Because we need the union to be countable, it is not adequate to simply say "Each point is in an open interval, which stays inside the open set."

Hint: We need a way to capture the intervals, such that when we "count" one interval we don't also count it again at some other point. This can be accomplished by using an equivalence relation, since equivalence relations afford a unique representation of each partition.

So define an equivalence relation on the open set, such that the cells of the corresponding partition are intervals disconnected from each other.

Exercise 6. Open Set Are Approximately Finite Unions of Intervals

Let $\varepsilon \in {\mathbb {R}}^{+}$ and U an open set. Show that there is a finite collection of open intervals, $I_{1},I_{2},...,I_{n}$, such that $\bigcup _{i=1}^{n}I_{i}\subseteq U$ and

$\lambda \left(U\smallsetminus \bigcup _{i=1}^{n}I_{i}\right)<\varepsilon$.

Exercise 7. Measurable sets are Approximately Finite Unions of Intervals

Let $\varepsilon \in {\mathbb {R}}^{+}$ and $E\in {\mathcal {M}}$. Show that there is a finite collection of open intervals, $I_{1},\dots ,I_{n}$, such that

$\lambda \left(E\Delta \bigcup _{i=1}^{n}I_{i}\right)<\varepsilon$

Hint: Approximate E by an open set, approximate the open set by intervals, and so on.