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

Lebesgue Measure. Definition

The Jordan measure has limitations. Tweak the Jordan measure to arrive at the Lebesgue measure.

Jordan measure on \( \mathbb{R}^d \)

Recap. The development of Jordan measure proceeded as follows:

First, one defines the notion of a box \( B \) and its volume \( |B| \).
Elementary sets
Define the notion of an elementary set \( E \) (a [something] of boxes) and define the elementary measure \( m(E) \) of such sets.
Jordan inner and outer measure
Define the inner and outer Jordan measures, \( m_{*,(J)}(F) \) and \( m^{*,(J)}(F) \), of an arbitrary bounded set \( F \subset \mathbb{R}^d \). These are limits of elementary measure of elementary sets that are either contained in (inner) \( F \) or contain (outer) \( F \).
Jordan measurability
If [something about \(F \) ], we say that \( F \) is Jordan measurable and call \( m(F) := m_{*,(J)}(F) = m^{*,(J)}(F) \) the Jordan measure of \( F \).

Jordan measure limitations

This concept of measure is perfectly satisfactory for any sets that are Jordan measurable. However, not all sets are Jordan measurable: the classic example is the [some set] and the [some related set]—both of these sets have Jordan outer measure 1 and Jordan inner measure 0.

More power to the Jordan outer measure

Trying to measure non-Jordan measurable sets leads us to develop the Lebesgue Measure.

Let's tinker with the Jordan outer measure to give it more power. The Jordan outer measure for a set \( F \subset \mathbb{R}^d \) is defined as:

\[ m^{*,(J)}(F) := \inf_{F \subseteq E; \, E \text{ is elementary}} m(E) \]

Jordan outer measure

As an elementary set is made up of boxes, we can rewrite the Jordan outer measure definition as:
\[ \begin{aligned} m^{*,(J)}(F) &:= \inf_{F \subseteq B_1 \cup ... \cup B_k;\, B_1, ..., B_k \text{ are boxes}} |B_1| + ... + |B_k| \\ &= \inf_{F \subseteq \cup_{n=1}^{k}B_n;\, B_1, ..., B_k \text{ are boxes}} \sum_{n=1}^{k} |B_n| \end{aligned} \]

Focus on the bit under then infimum. In words, the Jordan measure is the infimal cost (or volume) required to cover \( F \) by [a something of boxes].

Lebesgue outer measure

The tweak: allow a countable union of boxes instead of just a finite union. This is the Lebesgue outer measure of \( F \):

\[ m^{*}(F) := \inf_{F \subseteq \cup_{n=1}^{\infty}B_n; \, B_1, ..., \text{ are boxes}} \sum_{n=1}^{\infty} |B_n| \]

Can you spot the tiny tweak?

In words, the Lebesgue outer measure is the infimal cost (or volume) required to cover \( F \) by [a something of boxes].