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:

Boxes
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].