Support (measure theory)
In mathematics, the support (sometimes topological support or spectrum) of a measure μ on a measurable topological space (X, Borel(X)) is a precise notion of where in the space X the measure "lives". It is defined to be the largest (closed) subset of X for which every open neighbourhood of every point of the set has positive measure.
A (non-negative) measure μ on a measurable space (X, Σ) is really a function μ : Σ → [0, +∞]. Therefore, in terms of the usual definition of support, the support of μ is a subset of the σ-algebra Σ:
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. What we really want to know is where in the space X the measure μ is non-zero. Consider two examples:
In light of these two examples, we can reject the following candidate definitions in favour of the one in the next section:
However, the idea of "local strict positivity" is not too far from a workable definition:
Let (X, T) be a topological space; let B(T) denote the Borel σ-algebra on X, i.e. the smallest sigma algebra on X that contains all open sets U ∈ T. Let μ be a measure on (X, B(T)). Then the support (or spectrum) of μ is defined as the set of all points x in X for which every open neighbourhood Nx of x has positive measure:
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