# 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:

#### Jordan outer measure

As an elementary set is made up of boxes, we can rewrite the Jordan outer measure definition as: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
\( F \):

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].