Support (measure theory)

In mathematics, the support (sometimes topological support or spectrum) of a measure on a measurable topological space is a precise notion of where in the space the measure "lives". It is defined to be the largest (closed) subset of for which every open neighbourhood of every point of the set has positive measure.

Motivation

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A (non-negative) measure   on a measurable space   is really a function   Therefore, in terms of the usual definition of support, the support of   is a subset of the σ-algebra     where the overbar denotes set closure. However, this definition is somewhat unsatisfactory: we use the notion of closure, but we do not even have a topology on   What we really want to know is where in the space   the measure   is non-zero. Consider two examples:

  1. Lebesgue measure   on the real line   It seems clear that   "lives on" the whole of the real line.
  2. A Dirac measure   at some point   Again, intuition suggests that the measure   "lives at" the point   and nowhere else.

In light of these two examples, we can reject the following candidate definitions in favour of the one in the next section:

  1. We could remove the points where   is zero, and take the support to be the remainder   This might work for the Dirac measure   but it would definitely not work for   since the Lebesgue measure of any singleton is zero, this definition would give   empty support.
  2. By comparison with the notion of strict positivity of measures, we could take the support to be the set of all points with a neighbourhood of positive measure:   (or the closure of this). It is also too simplistic: by taking   for all points   this would make the support of every measure except the zero measure the whole of  

However, the idea of "local strict positivity" is not too far from a workable definition.

Definition

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Let   be a topological space; let   denote the Borel σ-algebra on   i.e. the smallest sigma algebra on   that contains all open sets   Let   be a measure on   Then the support (or spectrum) of   is defined as the set of all points   in   for which every open neighbourhood   of   has positive measure:  

Some authors prefer to take the closure of the above set. However, this is not necessary: see "Properties" below.

An equivalent definition of support is as the largest   (with respect to inclusion) such that every open set which has non-empty intersection with   has positive measure, i.e. the largest   such that:  

Signed and complex measures

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This definition can be extended to signed and complex measures. Suppose that   is a signed measure. Use the Hahn decomposition theorem to write   where   are both non-negative measures. Then the support of   is defined to be  

Similarly, if   is a complex measure, the support of   is defined to be the union of the supports of its real and imaginary parts.

Properties

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  holds.

A measure   on   is strictly positive if and only if it has support   If   is strictly positive and   is arbitrary, then any open neighbourhood of   since it is an open set, has positive measure; hence,   so   Conversely, if   then every non-empty open set (being an open neighbourhood of some point in its interior, which is also a point of the support) has positive measure; hence,   is strictly positive. The support of a measure is closed in  as its complement is the union of the open sets of measure  

In general the support of a nonzero measure may be empty: see the examples below. However, if   is a Hausdorff topological space and   is a Radon measure, a Borel set   outside the support has measure zero:   The converse is true if   is open, but it is not true in general: it fails if there exists a point   such that   (e.g. Lebesgue measure). Thus, one does not need to "integrate outside the support": for any measurable function   or    

The concept of support of a measure and that of spectrum of a self-adjoint linear operator on a Hilbert space are closely related. Indeed, if   is a regular Borel measure on the line   then the multiplication operator   is self-adjoint on its natural domain   and its spectrum coincides with the essential range of the identity function   which is precisely the support of  [1]

Examples

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Lebesgue measure

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In the case of Lebesgue measure   on the real line   consider an arbitrary point   Then any open neighbourhood   of   must contain some open interval   for some   This interval has Lebesgue measure   so   Since   was arbitrary,  

Dirac measure

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In the case of Dirac measure   let   and consider two cases:

  1. if   then every open neighbourhood   of   contains   so  
  2. on the other hand, if   then there exists a sufficiently small open ball   around   that does not contain   so  

We conclude that   is the closure of the singleton set   which is   itself.

In fact, a measure   on the real line is a Dirac measure   for some point   if and only if the support of   is the singleton set   Consequently, Dirac measure on the real line is the unique measure with zero variance (provided that the measure has variance at all).

A uniform distribution

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Consider the measure   on the real line   defined by   i.e. a uniform measure on the open interval   A similar argument to the Dirac measure example shows that   Note that the boundary points 0 and 1 lie in the support: any open set containing 0 (or 1) contains an open interval about 0 (or 1), which must intersect   and so must have positive  -measure.

A nontrivial measure whose support is empty

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The space of all countable ordinals with the topology generated by "open intervals" is a locally compact Hausdorff space. The measure ("Dieudonné measure") that assigns measure 1 to Borel sets containing an unbounded closed subset and assigns 0 to other Borel sets is a Borel probability measure whose support is empty.

A nontrivial measure whose support has measure zero

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On a compact Hausdorff space the support of a non-zero measure is always non-empty, but may have measure   An example of this is given by adding the first uncountable ordinal   to the previous example: the support of the measure is the single point   which has measure  

References

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  1. ^ Mathematical methods in Quantum Mechanics with applications to Schrödinger Operators
  • Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. ETH Zürich, Birkhäuser Verlag, Basel. ISBN 3-7643-2428-7.{{cite book}}: CS1 maint: multiple names: authors list (link)
  • Parthasarathy, K. R. (2005). Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI. p. xii+276. ISBN 0-8218-3889-X. MR2169627 (See chapter 2, section 2.)
  • Teschl, Gerald (2009). Mathematical methods in Quantum Mechanics with applications to Schrödinger Operators. AMS.(See chapter 3, section 2)