nLab
Lebesgue measure

Idea

The Lebesgue measure is the usual measure on the real line (or on any Cartesian space). It may be characterised as the Radon measure which is translation invariant and assigns measure $1$ to the unit interval (or unit cube). It generalises (up to a scalar constant) to Haar measure on any locally compact topological group.

History

The Lebesgue measure’s origins can be traced to the broader theory of Lebesgue integration. The original purpose of the latter, in broad terms, was to expand the class of integrable functions in order to give meaning to functions that are not Riemann integrable. In order to accomplish this, the basic properties of the concept of the length of an interval must be understood. This then leads to the need to fully define the concept of measure, particularly in relation to sets. We begin with a lemma and a corollary.

Lemma

Let I be an interval, $I=I_1\cup I_2\cup \cdots$ where $I_1,I_2,\dots$ are disjoint intervals. Then $\mid I\mid ={\sum }_{j=1}^{\mathrm{\infty }}\mid I_j\mid$ (interpreted so that ${\sum }_{j=1}^{\mathrm{\infty }}\mid I_j\mid$ can be $+\mathrm{\infty }$, either because one of the summands is $+\mathrm{\infty }$ or because the series diverges).

Corollary

If I is any interval, then

$$\mid I\mid =\inf \{\sum _{j=1}^{\mathrm{\infty }}\mid I_j\mid :I\subseteq \bigcup _{j=1}^{\mathrm{\infty }}{I}_j\}$${|I|} = inf \left\{\sum_{j=1}^{\infty} {|I_{j}|} : I \subseteq \bigcup_{j=1}^{\infty} I_{j}\right\}

where $\{I_j\}$ is any countable covering of I by intervals.

Now suppose $B$ is an arbitrary set of real numbers. In order for $B$ to be measurable, we must have $\mid B\mid \le {\sum }_{j=1}^{\mathrm{\infty }}\mid I_j\mid$ where ${\bigcup }_{j=1}^{\mathrm{\infty }}{I}_j$ is any countable covering of $B$ by intervals. We must also have $\mid B\mid \le {\bigcup }_{j=1}^{\mathrm{\infty }}\mid I_j\mid$ where the infinum is taken over all countable coverings of $B$ by intervals.

Definition

We define the Lebesgue outer measure? as follows:

$$\mid B\mid =\inf \{\sum _{j=1}^{\mathrm{\infty }}\mid I_j\mid :B\subseteq \bigcup _{j=1}^{\mathrm{\infty }}{I}_j\}.$${|B|} = inf \left\{\sum_{j=1}^{\infty} {|I_{j}|} : B \subseteq \bigcup_{j=1}^{\infty} I_{j}\right\}.

The set $B$ is Lebesgue measurable if

$$\mid B\mid =\mid A\cap B\mid +\mid A\setminus B\mid$${|B|} = {|A \cap B|} + {|A \setminus B|}

holds for every set $A$. Restricting to these sets, Lebesgue outer measure becomes an honest measure.

Note that once the Lebesgue measure is known for open sets, the outer regularity property allows us to find it for all Borel sets (but also rather more sets).

References

• Wikipedia.
• nForum

For a very intuitive and readable derivation, see:

• R. Strichartz, The Way of Analysis, Jones & Bartlett, 2000.