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Math and science::Analysis::Tao, measure::02. Lebesgue measure

Outer regularity. Theorem.

This property connects the outer Lebesgue measure of an arbitrary set \( E \subset R^d \) to the outer Lebesgue measure of open sets which contain \( E \).

Outer regularity. Theorem.

Let \( E \subset \mathbb{R}^d \) be an arbitrary set. Then one has:

\[ m^*(E) = \inf_{E \subset U, U \text{is open} } m^*(U). \]

The proof can be intuitively grasped by noticing that \( m^*(E) \) can be used to specify an open cover of \( E \) with measure arbitrarily close to \( m^*(E) \).

The prove is illustrative of the \( \frac{\varepsilon}{2^n} \) trick.

Proof

From monotonicity of outer Lebesgue measure, we have:

\[ m^*(E) \le \inf_{E \subset U, U \text{is open} } m^*(U). \]

So we are left to show the reverse:

\[ \inf_{E \subset U, U \text{is open} } m^*(U) \le m^*(E). \]

Reading this second less-equal as "not greater than" can motivate intuition.

Tao points out specifically that the inequality is trivial if \( m^*(E) \) is infinite, and we can assume that \( m^*(E) \) is finite.

What follows is a exemplary use of the \( \frac{\varepsilon}{2^n} \) trick.

Let \( \varepsilon > 0 \) be a real. The definition of outer Lebesgue measure affords us the ability to assert the existance of a countable family of boxes covering \( E \) such that it's outer measure is slightly greater than the infimum:

\[ \sum_{i=1}^{\infty} |B_i| \le m^*(E) + \varepsilon \]

We can enlarge each box \( B_i \) to become an open box \( B'_i \) such that \( |B_i'| \le |B_i| + \frac{\varepsilon}{2^i} \). The union of these open boxes is also open and also contains \( E \). In particular, we have:

\[ \sum_{i=1}^{\infty}|B'_i| \le m^*(E) + \varepsilon + \sum_{i=1}^{\infty} \frac{\varepsilon}{2^i} = m^*(E) + 2\varepsilon. \]

Applying subadditivity (the outer measure of the union is less than the sum of the outer measures) we have:

\[ m^*\left( \bigcup_{i=1}^{\infty} B'_i \right) \le m^*(E) + 2\varepsilon. \]

Denote this union as \( B' = \cup_{i=1}^{\infty}B'_i \). We have found an open set \( B' \) that contains \( E \) and has outer measure \( m^*(B') \le m^*(E) + 2\varepsilon \). The infimum of measure over open covers of \( E \) must be less than this one instance of an open cover:

\[ \inf_{E \subset U, U \text{is open}} m^*(U) \le m^*(B') \le m^*(E) + 2\varepsilon. \]

As \( \varepsilon \) was arbitrary, we have:

\[ \inf_{E \subset U, U \text{is open} } m^*(U) \le m^*(E), \]
and combined with our initial subadditivity claim, we have:
\[ \inf_{E \subset U, U \text{is open}} m^*(U) =  m^*(E). \]

Source

p25