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 $R^{d}$

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 $F \subset R^{d}$ . 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 $F \subset R^{d}$ is defined as:

m^{*, (J)} (F) := inf_{F \subseteq E; E 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} are boxes} | B_{1} | + . . . + | B_{k} | \\ = inf_{F \subseteq \cup_{n = 1}^{k} B_{n}; B_{1}, . . ., B_{k} 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}, . . ., 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].