In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, two positive (or signed

Signed measure

In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. Some authors may call it a charge, by analogy with electric charge, which is a familiar distribution that takes on positive and negative values.-Definition:There are two slightly...

 or complex

Complex measure

In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size is a complex number.-Definition:...

) measures μ and ν defined on a measurable space (Ω, Σ) are called singular if there exist two disjoint sets A and B in Σ whose union is Ω such that μ is zero on all measurable subsets of B while ν is zero on all measurable subsets of A. This is denoted by

A refined form of Lebesgue's decomposition theorem

Lebesgue's decomposition theorem

In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem is a theorem which states that given \mu and \nu two σ-finite signed measures on a measurable space , there exist two σ-finite signed measures \nu_0 and \nu_1 such that:* \nu=\nu_0+\nu_1\, * \nu_0\ll\mu *...

 decomposes a singular measure into a singular continuous measure and a discrete measure

Discrete measure

In mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure if its support is at most a countable set. Note that the support need not be a discrete set...

. See below for examples.

Examples on Rⁿ

As a particular case, a measure defined on the Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 Rⁿ is called singular, if it is singular in respect to the Lebesgue measure

Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

 on this space. For example, the Dirac delta function

Dirac delta function

The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

 is a singular measure.

Example. A discrete measure

Discrete measure

In mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure if its support is at most a countable set. Note that the support need not be a discrete set...

.

The Heaviside step function

Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....

 on the real line

Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

,



has the Dirac delta distribution

Dirac delta function

The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

  as its distributional derivative. This is a measure on the real line, a "point mass" at 0. However, the Dirac measure  is not absolutely continuous with respect to Lebesgue measure , nor is absolutely continuous with respect to : but ; if is any open set

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 not containing 0, then but .

Example. A singular continuous measure.

The Cantor distribution has a cumulative distribution function

Cumulative distribution function

In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

 that is continuous but not absolutely continuous, and indeed its absolutely continuous part is zero: it is singular continuous.

See also

• Lebesgue's decomposition theorem
  Lebesgue's decomposition theorem
  In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem is a theorem which states that given \mu and \nu two σ-finite signed measures on a measurable space , there exist two σ-finite signed measures \nu_0 and \nu_1 such that:* \nu=\nu_0+\nu_1\, * \nu_0\ll\mu *...
  
  
• Absolutely continuous
• Singular distribution
  Singular distribution
  In probability, a singular distribution is a probability distribution concentrated on a set of Lebesgue measure zero, where the probability of each point in that set is zero. These distributions are sometimes called singular continuous distributions...