Support (measure theory)

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The support of a measure

What it means
- The support of a measure μ on a topological space X is a (usually closed) set inside X that tells you where the measure “lives.” Intuitively, it is the part of X where every small neighborhood has positive μ-mass.

A simple definition
- If X has a topology and μ is a measure on X, the support is the set
supp(μ) = { x in X : for every open neighborhood U of x, μ(U) > 0 }.
- Equivalently, the support is the largest closed set C such that every open set that meets C has positive μ-measure:
C is closed and whenever U is open with U ∩ C ≠ ∅, we have μ(U ∩ C) > 0.

Key facts
- supp(μ) is always closed.
- If μ is a nonzero measure, the support is nonempty in many common spaces (though in some exotic spaces it can be empty).
- A measure is strictly positive exactly when its support is all of X.

For signed and complex measures
- If μ is a signed measure, write μ = μ+ − μ− with μ+ and μ− nonnegative. Then
supp(μ) = supp(μ+) ∪ supp(μ−).
- If μ is a complex measure, its support is the union of the supports of its real and imaginary parts.

When the support is full or empty
- supp(μ) = X if and only if μ is strictly positive (every open set has positive mass).
- Some measures can have support smaller than X; for example, Dirac measure δp has supp(δp) = {p}.

Examples
- Lebesgue measure on the real line: supp = R.
- Dirac measure at p: supp = {p}.
- Uniform measure on the interval (0,1): supp = [0,1] (the endpoints are included because any open set around 0 or 1 intersects (0,1) with positive mass).
- A measure that assigns mass 1 to Borel sets that contain some unbounded closed subset and 0 to other Borel sets can have an empty support in some spaces (an unusual example).

Why it matters
- The support tells you where the measure is concentrated. If you integrate a function, you can often restrict the integral to the support:
∫ f dμ = ∫_{supp(μ)} f dμ.
- The concept connects to other areas, such as the spectrum of certain operators in analysis.

A note on a common relation
- For a regular Borel measure on the real line, the spectrum of a natural multiplication operator by x in L2 is exactly the support of the measure.

In short
- The support is the set of points where every small neighborhood carries positive mass under μ. It is a useful, closed description of where a measure “lives,” with simple, concrete examples to keep in mind.