nLab Lebesgue measure

Lebesgue measure

Lebesgue measure


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 11 to the unit interval (or unit cube). It generalises (up to a scalar constant) to Haar measure on any locally compact topological group.


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.


Let I be an interval, I=I 1I 2I = I_{1} \cup I_{2} \cup \cdots where I 1,I 2,I_{1}, I_{2}, \ldots are disjoint intervals. Then |I|= j=1 |I j|{|I|} = \sum_{j=1}^{\infty} {|I_{j}|} (interpreted so that j=1 |I j|\sum_{j=1}^{\infty} {|I_{j}|} can be ++\infty, either because one of the summands is ++\infty or because the series diverges).


If I is any interval, then

|I|=inf{ j=1 |I j|:I j=1 I j} {|I|} = inf \left\{\sum_{j=1}^{\infty} {|I_{j}|} : I \subseteq \bigcup_{j=1}^{\infty} I_{j}\right\}

where {I j}\{I_{j}\} is any countable covering of I by intervals.

Now suppose BB is an arbitrary set of real numbers. In order for BB to be measurable, we must have |B| j=1 |I j|{|B|} \leq \sum_{j=1}^{\infty} {|I_{j}|} where j=1 I j\bigcup_{j=1}^{\infty} I_{j} is any countable covering of BB by intervals. We must also have |B| j=1 |I j|{|B|} \leq \bigcup_{j=1}^{\infty} {|I_{j}|} where the infinum is taken over all countable coverings of BB by intervals.


We define the Lebesgue outer measure? as follows:

|B|=inf{ j=1 |I j|:B j=1 I j}. {|B|} = inf \left\{\sum_{j=1}^{\infty} {|I_{j}|} : B \subseteq \bigcup_{j=1}^{\infty} I_{j}\right\}.

The set BB is Lebesgue measurable if

|A|=|AB|+|AB| {|A|} = {|A \cap B|} + {|A \setminus B|}

holds for every set AA. 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).

See also


Named after Henri Lebesgue.

See also

For a very intuitive and readable derivation, see:

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

Last revised on February 11, 2024 at 12:33:03. See the history of this page for a list of all contributions to it.