In mathematics, the support (sometimes topological support or spectrum) of a measure μ {\displaystyle \mu } on a measurable topological space ( X , Borel ( X ) ) {\displaystyle (X,\operatorname {Borel} (X))} is a precise notion of where in the space X {\displaystyle X} the measure "lives". It is defined to be the largest (closed) subset of X {\displaystyle X} for which every open neighbourhood of every point of the set has positive measure.
Motivation A (non-negative) measure μ {\displaystyle \mu } on a measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} is really a function μ : Σ → [ 0 , + ∞ ] . {\displaystyle \mu :\Sigma \to [0,+\infty ].} Therefore, in terms of the usual definition of support, the support of μ {\displaystyle \mu } is a subset of the σ-algebra Σ : {\displaystyle \Sigma :}
supp ( μ ) := { A ∈ Σ | μ ( A ) ≠ 0 } ¯ , {\displaystyle \operatorname {supp} (\mu ):={\overline {\{A\in \Sigma \,\vert \,\mu (A)\neq 0\}}},}
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 Σ . {\displaystyle \Sigma .} What we really want to know is where in the space X {\displaystyle X} the measure μ {\displaystyle \mu } is non-zero. Consider two examples:
Lebesgue measure λ {\displaystyle \lambda } on the real line R . {\displaystyle \mathbb {R} .} It seems clear that λ {\displaystyle \lambda } "lives on" the whole of the real line. A Dirac measure δ p {\displaystyle \delta _{p}} at some point p ∈ R . {\displaystyle p\in \mathbb {R} .} Again, intuition suggests that the measure δ p {\displaystyle \delta _{p}} "lives at" the point p , {\displaystyle p,} 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:
We could remove the points where μ {\displaystyle \mu } is zero, and take the support to be the remainder X ∖ { x ∈ X ∣ μ ( { x } ) = 0 } . {\displaystyle X\setminus \{x\in X\mid \mu (\{x\})=0\}.} This might work for the Dirac measure δ p , {\displaystyle \delta _{p},} but it would definitely not work for λ : {\displaystyle \lambda :} since the Lebesgue measure of any singleton is zero, this definition would give λ {\displaystyle \lambda } empty support. 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: { x ∈ X ∣ ∃ N x open such that ( x ∈ N x and μ ( N x ) > 0 ) } {\displaystyle \{x\in X\mid \exists N_{x}{\text{ open}}{\text{ such that }}(x\in N_{x}{\text{ and }}\mu (N_{x})>0)\}} (or the closure of this). It is also too simplistic: by taking N x = X {\displaystyle N_{x}=X} for all points x ∈ X , {\displaystyle x\in X,} this would make the support of every measure except the zero measure the whole of X . {\displaystyle X.}
However, the idea of "local strict positivity" is not too far from a workable definition.
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