The theorem states that if (\Omega,\Sigma) is a
measurable space and \mu and \nu are
σ-finite signed measures on \Sigma, then there exist two uniquely determined σ-finite signed measures \nu_0 and \nu_1 such that: • \nu=\nu_0+\nu_1\, • \nu_0\ll\mu (that is, \nu_0 is
absolutely continuous with respect to \mu) • \nu_1\perp\mu (that is, \nu_1 and \mu are
singular).
Refinement Lebesgue's decomposition theorem can be refined in a number of ways. First, as the
Lebesgue–Radon–Nikodym theorem. That is, let (\Omega,\Sigma) be a measure space, \mu a
σ-finite positive measure on \Sigma and \lambda a
complex measure on \Sigma. • There is a unique pair of complex measures on \Sigma such that \lambda = \lambda_a + \lambda_s, \quad \lambda_a \ll \mu, \quad \lambda_s \perp \mu. If \lambda is positive and finite, then so are \lambda_a and \lambda_s. • There is a unique h \in L^1(\mu) such that \lambda_a (E) = \int_E h d\mu, \quad E \in \Sigma. The first assertion follows from the Lebesgue decomposition, the second is known as the
Radon–Nikodym theorem. That is, the function h is a Radon–Nikodym derivative that can be expressed as h = \frac{d\lambda_a}{d\mu}. An alternative refinement is that of the decomposition of a regular
Borel measure \nu = \nu_{ac} + \nu_{sc} + \nu_{pp}, where • \nu_{ac} \ll \mu is the
absolutely continuous part • \nu_{sc} \perp \mu is the
singular continuous part • \nu_{pp} is the
pure point part (a
discrete measure). The 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==