Lebesgue's decomposition theorem explained

In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures

\mu

and

\nu

on a measurable space

(\Omega,\Sigma),

there exist two σ-finite signed measures

\nu0

and

\nu1

such that:

\nu=\nu0+\nu1

\nu0\ll\mu

(that is,

\nu0

is absolutely continuous with respect to

\mu

)

\nu1\perp\mu

(that is,

\nu1

and

\mu

are singular).

These two measures are uniquely determined by

\mu

and

\nu.

Refinement

Lebesgue's decomposition theorem can be refined in a number of ways.

First, the decomposition of a regular Borel measure on the real line can be refined:

\nu=\nucont+\nusing+\nupp

where

Second, absolutely continuous measures are classified by the Radon–Nikodym theorem, and discrete measures are easily understood. Hence (singular continuous measures aside), Lebesgue decomposition gives a very explicit description of measures. The Cantor measure (the probability measure on the real line whose cumulative distribution function is the Cantor function) is an example of a singular continuous measure.

Related concepts

Lévy–Itō decomposition

The analogous decomposition for a stochastic processes is the Lévy–Itō decomposition: given a Lévy process X, it can be decomposed as a sum of three independent Lévy processes

X=X(1)+X(2)+X(3)

where:

X(1)

is a Brownian motion with drift, corresponding to the absolutely continuous part;

X(2)

is a compound Poisson process, corresponding to the pure point part;

X(3)

is a square integrable pure jump martingale that almost surely has a countable number of jumps on a finite interval, corresponding to the singular continuous part.

See also